Your FREE GRE Mathematics Subject Test Practice Questions 2026 – 150+ Q&A

Calculus (74)

1. What is the derivative of $f(x) = e^{2x} \sin(x)$?

   • A.$e^{2x} \cos(x)$
   • B.$2e^{2x} \sin(x) + e^{2x} \cos(x)$
   • C.$e^{2x} \sin(x) + e^{2x} \cos(x)$
   • D.$2e^{2x} \cos(x) - e^{2x} \sin(x)$
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   Correct answer: $2e^{2x} \sin(x) + e^{2x} \cos(x)$

   Correct answer: $2e^{2x} \sin(x) + e^{2x} \cos(x)$. Explanation: Use the product rule for differentiation.

2. Find the integral of $\int \frac{1}{x \ln(x)} dx$.

   • A.$\ln | \ln(x) | + C$
   • B.$\ln |x| + C$
   • C.$\frac{1}{\ln(x)} + C$
   • D.$\frac{x}{\ln(x)} + C$
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   Correct answer: $\ln | \ln(x) | + C$

   Correct answer: $\ln | \ln(x) | + C$. Explanation: Use substitution, setting $u = \ln(x)$.

3. What is the area enclosed by the curve $y = x^2$ and the lines $x = -1, x = 2,$ and $y = 0$?

   • A.3 square units
   • B.4.5 square units
   • C.9 square units
   • D.5.5 square units
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   Correct answer: 3 square units

   Correct answer: 3 square units. Explanation: Calculate the definite integral of $x^2$ from -1 to 2.

4. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$.

   • A.Converges
   • B.Diverges
   • C.Converges conditionally
   • D.Cannot be determined
   Show answerHide answer

   Correct answer: Converges

   Correct answer: Converges. Explanation: Use the p-series test where $p = 2 > 1$, indicating convergence.

5. If $f(x) = \ln(x)$, what is $f''(e)$?

   • A.$\frac{1}{e}$
   • B.$-\frac{1}{e}$
   • C.$\frac{1}{e^2}$
   • D.$-\frac{1}{e^2}$
   Show answerHide answer

   Correct answer: $-\frac{1}{e^2}$

   Correct answer: $-\frac{1}{e^2}$. Explanation: Calculate the second derivative of $\ln(x)$ and evaluate at $x = e$.

6. Calculate the derivative of $f(x) = \sqrt{x}(x^2 + 1)$.

   • A.$\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$
   • B.$\frac{1}{2\sqrt{x}} + 2x\sqrt{x}$
   • C.$\frac{3}{2}x\sqrt{x}$
   • D.$\frac{1}{2}x^{-\frac{1}{2}} + 2x^\frac{3}{2}$
   Show answerHide answer

   Correct answer: $\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$

   Correct answer: $\frac{1}{2\sqrt{x}}(x^2 + 1) + 2x\sqrt{x}$. Explanation: Apply the product rule and simplify.

7. Evaluate the definite integral $\int_{0}^{2} (3x^2 - 2x) dx$.

   • A.4
   • B.6
   • C.8
   • D.10
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   Correct answer: 4

   Correct answer: 4. Explanation: Find the antiderivative $x^3 - x^2$ and apply the Fundamental Theorem of Calculus: $(8 - 4) - 0 = 4$.

8. Find the derivative of $g(t) = t^4 \ln(t)$.

   • A.$4t^3 \ln(t)$
   • B.$4t^3 \ln(t) + t^3$
   • C.$4t^3 + t^3 \ln(t)$
   • D.$t^4 + 4t^3 \ln(t)$
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   Correct answer: $4t^3 \ln(t) + t^3$

   Correct answer: $4t^3 \ln(t) + t^3$. Explanation: Use the product rule for differentiation.

9. Evaluate the integral: $\int (3x^2 - 2x + 1) dx$.

   • A.$x^3 - x^2 + x + C$
   • B.$x^3 - 2x^2 + x + C$
   • C.$x^3 - x^2 + 2x + C$
   • D.$x^3 + x^2 + x + C$
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   Correct answer: $x^3 - x^2 + x + C$

   Correct answer: $x^3 - x^2 + x + C$. Explanation: Integrate each term separately.

10. Determine the limit: $\lim_{x \to \infty} \frac{4x^3 - x}{2x^3 + 5}$.

    • A.0
    • B.1
    • C.$\frac{4}{2}$
    • D.$\infty$
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    Correct answer: $\frac{4}{2}$

    Correct answer: $\frac{4}{2}$. Explanation: Divide each term by $x^3$ and take the limit.

11. Find the area under the curve $y = e^{2x}$ from $x = 0$ to $x = 1$.

    • A.$e^2 - 1$
    • B.$\frac{e^2 - 1}{2}$
    • C.$e^2$
    • D.$\frac{e^2}{2}$
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    Correct answer: $\frac{e^2 - 1}{2}$

    Correct answer: $\frac{e^2 - 1}{2}$. Explanation: Evaluate the definite integral of the function.

12. Evaluate: $\lim_{x \to 0} \frac{\sin(3x)}{x}$.

    • A.0
    • B.1
    • C.3
    • D.Undefined
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    Correct answer: 3

    Correct answer: 3. Explanation: Apply the standard limit $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ with a change of variable.

13. What is the derivative of $f(x) = x^5 \sin(x^2)$?

    • A.$5x^4 \sin(x^2) + 2x^6 \cos(x^2)$
    • B.$5x^4 \sin(x^2) + x^6 \cos(x^2)$
    • C.$5x^4 \cos(x^2)$
    • D.$x^5 \cos(x^2)$
    Show answerHide answer

    Correct answer: $5x^4 \sin(x^2) + 2x^6 \cos(x^2)$

    Correct answer: $5x^4 \sin(x^2) + 2x^6 \cos(x^2)$. Explanation: Apply the product rule and the chain rule.

14. Calculate the derivative of $f(x) = \frac{1}{x^2 + x + 1}$.

    • A.$\frac{-2x - 1}{(x^2 + x + 1)^2}$
    • B.$\frac{-2x + 1}{(x^2 + x + 1)^2}$
    • C.$\frac{2x - 1}{(x^2 + x + 1)^2}$
    • D.$\frac{2x + 1}{(x^2 + x + 1)^2}$
    Show answerHide answer

    Correct answer: $\frac{-2x - 1}{(x^2 + x + 1)^2}$

    Correct answer: $\frac{-2x - 1}{(x^2 + x + 1)^2}$. Explanation: Apply the quotient rule to find the derivative.

15. Evaluate the limit: $\lim_{x \to 0} \frac{e^{x} - 1}{x}$.

    • A.0
    • B.1
    • C.$e$
    • D.Undefined
    Show answerHide answer

    Correct answer: 1

    Correct answer: 1. Explanation: This is a standard limit for exponential functions.

16. Find the indefinite integral: $\int x \cos(x^2) dx$.

    • A.$\frac{1}{2}\sin(x^2) + C$
    • B.$\sin(x^2) + C$
    • C.$\cos(x^2) + C$
    • D.$\frac{1}{2}\cos(x^2) + C$
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    Correct answer: $\frac{1}{2}\sin(x^2) + C$

    Correct answer: $\frac{1}{2}\sin(x^2) + C$. Explanation: Use substitution with $u = x^2$.

17. Compute the area between the curves $y = x^2$ and $y = x$ for $0 \le x \le 1$.

    • A.$\frac{1}{6}$
    • B.$\frac{1}{3}$
    • C.$\frac{1}{2}$
    • D.$\frac{2}{3}$
    Show answerHide answer

    Correct answer: $\frac{1}{6}$

    Correct answer: $\frac{1}{6}$. Explanation: Subtract the integrals of the two functions over the given interval.

18. Determine the convergence or divergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3}$.

    • A.Converges
    • B.Diverges
    • C.Converges conditionally
    • D.Cannot be determined
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    Correct answer: Converges

    Correct answer: Converges. Explanation: The series converges absolutely: the terms $\frac{1}{n^3}$ form a convergent p-series with $p = 3 > 1$, so the alternating series converges. 'Converges conditionally' is incorrect because the absolute series already converges.

19. What is the maximum value of the function $f(x) = -x^2 + 4x - 3$ on the interval [0, 3]?

    • A.1
    • B.2
    • C.3
    • D.4
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    Correct answer: 1

    Correct answer: 1. Explanation: Find the vertex of the parabola or use calculus to find the maximum.

20. Evaluate the integral: $\int \frac{dx}{x\sqrt{x^2 - 1}}$.

    • A.$\ln |x + \sqrt{x^2 - 1}| + C$
    • B.$\arcsin \frac{1}{x} + C$
    • C.$\operatorname{arcsec} x + C$
    • D.$\ln |x^2 - \sqrt{x^2 - 1}| + C$
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    Correct answer: $\operatorname{arcsec} x + C$

    Correct answer: $\operatorname{arcsec} x + C$. Explanation: Use substitution and trigonometric identities.

21. Find the derivative of $g(y) = \arctan(\sqrt{y})$.

    • A.$\frac{1}{2\sqrt{y}(1+y)}$
    • B.$\frac{1}{2\sqrt{y}(1+y^2)}$
    • C.$\frac{1}{1+y}$
    • D.$\frac{1}{1+y^2}$
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    Correct answer: $\frac{1}{2\sqrt{y}(1+y)}$

    Correct answer: $\frac{1}{2\sqrt{y}(1+y)}$. Explanation: Apply the chain rule and simplify.

22. Compute the limit: $\lim_{x \to \infty} \frac{\sqrt{x^2 + x} - x}{x}$.

    • A.0
    • B.$\frac{1}{2}$
    • C.1
    • D.Undefined
    Show answerHide answer

    Correct answer: 0

    Correct answer: 0. Explanation: Multiply by the conjugate and simplify.

23. Using integration by parts, what is the value of the integral of $x e^x$ with respect to $x$?

    • A.$x e^x + e^x + C$
    • B.$x e^x - e^x + C$
    • C.$\frac{x^2}{2} e^x + C$
    • D.$e^x(x + 1) + C$
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    Correct answer: $x e^x - e^x + C$

    The antiderivative is $x e^x - e^x + C$. Integration by parts uses the formula $\int u \, dv = u v - \int v \, du$; choosing $u = x$ (so $du = dx$) and $dv = e^x \, dx$ (so $v = e^x$) gives $x e^x - \int e^x \, dx = x e^x - e^x + C$. The tempting $x e^x + e^x + C$ reverses the sign of the second term.

24. Apply L'Hopital's rule to evaluate $\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$.

    • A.$\frac{1}{2}$
    • B.1
    • C.Does not exist
    • D.0
    Show answerHide answer

    Correct answer: $\frac{1}{2}$

    The limit equals $\frac{1}{2}$. The form is $\frac{0}{0}$, so differentiating top and bottom gives $\frac{e^x - 1}{2x}$, still $\frac{0}{0}$; a second application gives $\frac{e^x}{2}$, which approaches $\frac{1}{2}$ as $x \to 0$. Stopping after one application and concluding the limit is 0 misses the second indeterminate form.

25. The Mean Value Theorem guarantees a point $c$ in $(1, 4)$ for $f(x) = x^2$ where the instantaneous rate of change equals the average rate of change. What is $c$?

    • A.$\sqrt{5}$
    • B.3
    • C.2.5
    • D.2
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    Correct answer: 2.5

    The value is $c = 2.5$. The Mean Value Theorem requires $f'(c) = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5$; since $f'(x) = 2x$, setting $2c = 5$ gives $c = 2.5$, which lies in $(1, 4)$. The midpoint 2.5 of the interval coincidentally equals the answer here because $f$ is quadratic, but the theorem is satisfied by the derivative condition.

26. By the Fundamental Theorem of Calculus, if $F(x)$ is the integral from 0 to $x$ of $\cos(t^2) \, dt$, what is $F'(x)$?

    • A.$-\sin(x^2) \cdot 2x$
    • B.$\cos(x^2)$
    • C.$\frac{1}{2} \sin(x^2)$
    • D.$\cos(x^2) \cdot 2x$
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    Correct answer: $\cos(x^2)$

    $F'(x) = \cos(x^2)$. The Fundamental Theorem of Calculus (Part 1) states that the derivative of an integral with a constant lower limit and variable upper limit $x$ is simply the integrand evaluated at $x$. No chain-rule factor of $2x$ appears because the upper limit is $x$ itself, not $x^2$.

27. Using the chain rule, what is the derivative of $f(x) = (3x^2 + 1)^5$?

    • A.$30x(3x^2 + 1)^4$
    • B.$5(6x)^4$
    • C.$6x(3x^2 + 1)^4$
    • D.$5(3x^2 + 1)^4$
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    Correct answer: $30x(3x^2 + 1)^4$

    The derivative is $30x(3x^2 + 1)^4$. The chain rule multiplies the derivative of the outer power, $5(3x^2 + 1)^4$, by the derivative of the inner function $3x^2 + 1$, which is $6x$, giving $5 \cdot 6x = 30x$ times $(3x^2 + 1)^4$. Forgetting the inner derivative leaves the incorrect $5(3x^2 + 1)^4$.

28. By the alternating series test, which condition on the terms $b_n$ (all positive) must hold for the series $\sum (-1)^n b_n$ to converge?

    • A.$b_n$ must equal $\frac{1}{n}$ exactly
    • B.$b_n$ must be increasing
    • C.$b_n$ must approach 0 and be eventually decreasing
    • D.The sum of $b_n$ must converge
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    Correct answer: $b_n$ must approach 0 and be eventually decreasing

    The terms $b_n$ must approach 0 and be eventually monotonically decreasing. The alternating series test (Leibniz test) guarantees convergence when these two conditions hold, even if the series of absolute values diverges. Requiring the sum of $b_n$ to converge would describe absolute convergence, a strictly stronger condition than the test requires.

29. Green's theorem converts a line integral around a positively oriented simple closed curve $C$ into a double integral over the enclosed region $D$. For the field $F = (P, Q)$, the line integral of $P \, dx + Q \, dy$ equals the double integral over $D$ of which expression?

    • A.$\frac{\partial P}{\partial y} - \frac{\partial Q}{\partial x}$
    • B.$\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial x}$
    • C.$\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}$
    • D.$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$
    Show answerHide answer

    Correct answer: $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$

    The integrand is $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$. Green's theorem equates the counterclockwise circulation line integral of $P \, dx + Q \, dy$ to the double integral of the scalar curl, $Q_x - P_y$, over the enclosed region. Reversing this to $P_y - Q_x$ flips the sign and corresponds to clockwise orientation.

30. Using the limit comparison test with $b_n = \frac{1}{n^2}$, what can be concluded about the series $\sum_{n=1}^{\infty} \frac{n + 1}{n^3 + 5}$?

    • A.It converges
    • B.It diverges
    • C.It converges only conditionally
    • D.The test is inconclusive
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    Correct answer: It converges

    The series converges. The limit comparison test compares $a_n = \frac{n+1}{n^3+5}$ with $b_n = \frac{1}{n^2}$; the limit of $\frac{a_n}{b_n}$ equals the limit of $\frac{n^2(n+1)}{n^3+5} = 1$, a finite positive number. Since $\sum \frac{1}{n^2}$ is a convergent p-series ($p = 2 > 1$), the given series also converges.

31. What is the Maclaurin series for $\cos(x)$?

    • A.$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$
    • B.$1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots$
    • C.$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
    • D.$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
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    Correct answer: $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$

    The Maclaurin series for $\cos(x)$ is $1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$, containing only even powers of $x$ with alternating signs. A Maclaurin series is a Taylor series centered at 0; the series with only odd powers ($x - \frac{x^3}{3!} + \cdots$) is the expansion of $\sin(x)$, not cosine.

32. Using Lagrange multipliers to extremize $f(x, y) = x + y$ subject to the constraint $x^2 + y^2 = 8$, what is the maximum value of $f$?

    • A.$2\sqrt{2}$
    • B.2
    • C.4
    • D.$\sqrt{8}$
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    Correct answer: 4

    The maximum value is 4. Setting $\nabla f = \lambda \nabla g$ gives $1 = 2 \lambda x$ and $1 = 2 \lambda y$, so $x = y$; substituting into $x^2 + y^2 = 8$ yields $2x^2 = 8$, $x = 2$, so $x = y = 2$ and $f = 4$. The competing value $2\sqrt{2}$ is the distance $\sqrt{8}$, not the sum $x + y$.

33. For $f(x, y) = x^2 y + 3y^3$, what is the partial derivative of $f$ with respect to $y$?

    • A.$x^2 + 3y^2$
    • B.$2xy + 9y^2$
    • C.$2xy$
    • D.$x^2 + 9y^2$
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    Correct answer: $x^2 + 9y^2$

    The partial derivative with respect to $y$ is $x^2 + 9y^2$. When differentiating partially with respect to $y$, $x$ is held constant, so $x^2 y$ differentiates to $x^2$ and $3y^3$ differentiates to $9y^2$. The expression $2xy$ is the partial derivative with respect to $x$ of the first term, not with respect to $y$.

34. The integral test can be applied to $\sum_{n=1}^{\infty} \frac{1}{n (\ln n)^2}$ starting at $n = 2$. What does it conclude?

    • A.The series converges
    • B.The series oscillates
    • C.The integral test does not apply
    • D.The series diverges
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    Correct answer: The series converges

    The series converges. The integral test applies because $f(x) = \frac{1}{x (\ln x)^2}$ is positive, continuous, and decreasing for $x \ge 2$; the improper integral from 2 to infinity equals the limit of $-\frac{1}{\ln x}$, which converges to $\frac{1}{\ln 2}$. Since the integral converges, so does the series.

35. A spherical balloon is inflated so its volume increases at $36\pi$ cubic cm per second. Using related rates, how fast is the radius increasing when the radius is 3 cm? ($V = \frac{4}{3} \pi r^3$.)

    • A.1 cm/s
    • B.$4\pi$ cm/s
    • C.2 cm/s
    • D.3 cm/s
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    Correct answer: 1 cm/s

    The radius increases at 1 cm/s. Differentiating $V = \frac{4}{3} \pi r^3$ gives $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$; substituting $\frac{dV}{dt} = 36\pi$ and $r = 3$ gives $36\pi = 4 \pi (9) \frac{dr}{dt} = 36\pi \frac{dr}{dt}$, so $\frac{dr}{dt} = 1$. The related rates method links the rates through the chain rule applied to the volume formula.

36. Evaluate the line integral of the scalar field $f(x, y) = x$ along the curve $C$ parameterized by $x = t$, $y = t$ for $t$ from 0 to 1, with respect to arc length $ds$.

    • A.$\frac{\sqrt{2}}{2}$
    • B.1
    • C.$\frac{1}{2}$
    • D.$\sqrt{2}$
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    Correct answer: $\frac{\sqrt{2}}{2}$

    The line integral equals $\frac{\sqrt{2}}{2}$. With $x = t$ and $y = t$, $ds = \sqrt{(dx/dt)^2 + (dy/dt)^2} \, dt = \sqrt{2} \, dt$, and $f = t$, so the integral is $\int_0^1 t \sqrt{2} \, dt = \sqrt{2} \cdot \frac{1}{2} = \frac{\sqrt{2}}{2}$. Omitting the $\sqrt{2}$ arc-length factor gives the incorrect value $\frac{1}{2}$.

37. According to the epsilon-delta definition of a limit, the statement '$\lim_{x \to a} f(x) = L$' means: for every $\epsilon > 0$ there exists a $\delta > 0$ such that which condition holds?

    • A.If $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$
    • B.If $|f(x) - L| < \delta$ then $|x - a| < \epsilon$
    • C.If $x = a$ then $f(x) = L$
    • D.If $|x - a| < \epsilon$ then $|f(x) - L| < \delta$
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    Correct answer: If $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$

    The correct condition is: if $0 < |x - a| < \delta$ then $|f(x) - L| < \epsilon$. The epsilon-delta definition says that for any tolerance $\epsilon$ on the output, a positive $\delta$ on the input can be found so that all $x$ within $\delta$ of $a$ (but not equal to $a$) map to $f(x)$ within $\epsilon$ of $L$. Swapping the roles of $\epsilon$ and $\delta$ describes a different, incorrect statement.

38. What is the sum of the telescoping series $\sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n+1} \right)$?

    • A.0
    • B.1
    • C.$\frac{1}{2}$
    • D.Diverges
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    Correct answer: 1

    The sum is 1. In a telescoping series, successive terms cancel: the partial sum $S_N = 1 - \frac{1}{N+1}$, and as $N$ approaches infinity, $\frac{1}{N+1}$ approaches 0, leaving 1. Recognizing the cancellation pattern is the key technique rather than applying a convergence test.

39. Evaluate the double integral over the rectangle $0 \le x \le 2$, $0 \le y \le 3$ of the function $f(x, y) = x y$.

    • A.18
    • B.6
    • C.9
    • D.12
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    Correct answer: 9

    The double integral equals 9. Because the integrand separates as $x$ times $y$ over a rectangle, the value is $\left( \int_0^2 x \, dx \right) \left( \int_0^3 y \, dy \right) = 2 \cdot 4.5 = 9$. The value 18 would result from forgetting the factor of one-half in one of the single integrals.

40. Using the ratio test, what does the test conclude about the series $\sum_{n=1}^{\infty} \frac{n!}{10^n}$?

    • A.The ratio test is inconclusive
    • B.It converges
    • C.It converges conditionally
    • D.It diverges
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    Correct answer: It diverges

    The series diverges. The ratio test computes the limit of $\frac{a_{n+1}}{a_n} = \frac{(n+1)! / 10^{n+1}}{n! / 10^n} = \frac{n+1}{10}$, which approaches infinity. Since this limit exceeds 1, the ratio test guarantees divergence; the factorial grows far faster than the exponential $10^n$.

41. For what values of the common ratio $r$ does the geometric series $\sum_{n=0}^{\infty} r^n$ converge, and to what value?

    • A.All $r$, converging to $\frac{1}{1 + r}$
    • B.$|r| \le 1$, converging to $\frac{1}{1 - r}$
    • C.$r > 0$, converging to $\frac{r}{1 - r}$
    • D.$|r| < 1$, converging to $\frac{1}{1 - r}$
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    Correct answer: $|r| < 1$, converging to $\frac{1}{1 - r}$

    The geometric series converges if and only if $|r| < 1$, with sum $\frac{1}{1 - r}$. When the absolute value of the ratio is at least 1, the terms do not approach 0 and the series diverges. The endpoint $r = 1$ gives $1 + 1 + 1 + \cdots$, which diverges, so the inequality must be strict.

42. Apply the root test to the series $\sum_{n=1}^{\infty} \left( \frac{n}{2n+1} \right)^n$. What is the conclusion?

    • A.It converges to $\frac{1}{2}$
    • B.It converges
    • C.The root test is inconclusive
    • D.It diverges
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    Correct answer: It converges

    The series converges. The root test evaluates the limit of the $n$th root of $|a_n|$, which is the limit of $\frac{n}{2n+1} = \frac{1}{2}$. Because this limit $\frac{1}{2}$ is less than 1, the root test guarantees convergence. The value $\frac{1}{2}$ is the limit used by the test, not the sum of the series.

43. How do you find $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$?

    • A.The limit does not exist
    • B.Substitute directly to get 0
    • C.Factor and cancel to get 4
    • D.Apply the squeeze theorem to get 2
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    Correct answer: Factor and cancel to get 4

    Factoring and canceling gives the limit 4. The expression $\frac{x^2 - 4}{x - 2}$ factors as $\frac{(x - 2)(x + 2)}{x - 2}$; canceling the common factor leaves $x + 2$, which evaluates to 4 at $x = 2$. Direct substitution initially yields the indeterminate form $\frac{0}{0}$, so algebraic simplification is the correct technique.

44. What is the radius of convergence of the power series $\sum_{n=0}^{\infty} \frac{x^n}{n!}$?

    • A.$e$
    • B.1
    • C.0
    • D.Infinity
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    Correct answer: Infinity

    The radius of convergence is infinity. Applying the ratio test to $\frac{x^n}{n!}$ gives the limit of $\frac{|x| \, n!}{(n+1)!} = \frac{|x|}{n+1}$, which approaches 0 for every $x$. Since the limit is less than 1 for all real $x$, the series (which equals $e^x$) converges everywhere.

45. What is the difference between absolute convergence and conditional convergence of a series?

    • A.Absolute applies only to positive series; conditional only to negative series
    • B.Absolute means $\sum |a_n|$ converges; conditional means the series converges but $\sum |a_n|$ diverges
    • C.They are two names for the same property
    • D.Absolute means the series diverges; conditional means it converges
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    Correct answer: Absolute means $\sum |a_n|$ converges; conditional means the series converges but $\sum |a_n|$ diverges

    A series is absolutely convergent when the sum of the absolute values of its terms converges, whereas it is conditionally convergent when the series itself converges but the sum of absolute values diverges. The alternating harmonic series $\sum \frac{(-1)^n}{n}$ is the classic conditionally convergent example, since $\sum \frac{1}{n}$ diverges.

46. Using the direct comparison test, what is concluded about the series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 3n}$?

    • A.It diverges because each term is at least $\frac{1}{n}$
    • B.It converges only conditionally
    • C.The comparison test does not apply
    • D.It converges because each term is at most $\frac{1}{n^2}$
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    Correct answer: It converges because each term is at most $\frac{1}{n^2}$

    The series converges because $\frac{1}{n^2 + 3n} \le \frac{1}{n^2}$ for all $n \ge 1$, and $\sum \frac{1}{n^2}$ is a convergent p-series. The direct comparison test states that if terms of a positive series are dominated by the terms of a known convergent series, the given series also converges.

47. A function $f$ is differentiable at a point. Which statement about continuity and differentiability is true?

    • A.Differentiability at a point implies continuity there, but continuity does not imply differentiability
    • B.Continuity at a point implies differentiability there
    • C.Neither property implies the other
    • D.Differentiability implies the function is constant
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    Correct answer: Differentiability at a point implies continuity there, but continuity does not imply differentiability

    Differentiability at a point implies continuity at that point, but the converse fails. The absolute value function $|x|$ is continuous at $x = 0$ yet not differentiable there because its left and right derivatives differ. Thus continuity is a necessary but not sufficient condition for differentiability.

48. Determine whether the sequence $a_n = \frac{3n + 1}{n + 2}$ converges, and if so, to what limit.

    • A.Converges to 3
    • B.Converges to $\frac{1}{2}$
    • C.Diverges to infinity
    • D.Converges to 0
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    Correct answer: Converges to 3

    The sequence converges to 3. Dividing numerator and denominator by $n$ gives $\frac{3 + 1/n}{1 + 2/n}$; as $n$ approaches infinity, the $\frac{1}{n}$ and $\frac{2}{n}$ terms vanish, leaving $\frac{3}{1} = 3$. The ratio of the leading coefficients determines the limit when numerator and denominator have the same degree.

49. How can you tell whether the series $\sum_{n=1}^{\infty} \frac{1}{n}$ is convergent or divergent?

    • A.It converges because the terms approach $0$
    • B.It diverges to $-\infty$
    • C.It diverges; it is the harmonic series, a p-series with $p = 1$
    • D.It converges to $1$
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    Correct answer: It diverges; it is the harmonic series, a p-series with $p = 1$

    The harmonic series $\sum \frac{1}{n}$ diverges. It is a p-series with $p = 1$, and p-series diverge whenever $p \le 1$. The fact that the individual terms $\frac{1}{n}$ approach $0$ is necessary but not sufficient for convergence, which is a common misconception.

50. How do you find the Taylor series of a function $f$ about a point $a$?

    • A.$\sum_{n=0}^{\infty} f(a)^n (x - a)^n$
    • B.$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$
    • C.Differentiate $f$ once and multiply by $(x - a)$
    • D.$\sum_{n=0}^{\infty} \frac{f^{(n)}(x)}{n!} a^n$
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    Correct answer: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

    The Taylor series of $f$ about $a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$. Each coefficient is the $n$th derivative of $f$ evaluated at the center $a$, divided by $n!$. When the center $a$ is $0$, this reduces to the Maclaurin series.

51. For the scalar function $f(x, y) = x^2 + 3xy$, what is the gradient vector at the point $(1, 2)$?

    • A.$(8, 3)$
    • B.$(8, 6)$
    • C.$(2, 3)$
    • D.$(5, 6)$
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    Correct answer: $(8, 3)$

    The gradient is $(8, 3)$. The gradient in multivariable calculus is the vector of partial derivatives: $f_x = 2x + 3y$ and $f_y = 3x$; at $(1, 2)$ these give $2(1) + 3(2) = 8$ and $3(1) = 3$, so $\nabla f = (8, 3)$. The gradient points in the direction of steepest increase of $f$.

52. What is the Taylor series (about $0$) of the common function $\frac{1}{1 - x}$ for $|x| < 1$?

    • A.$1 + x + x^2 + x^3 + \cdots$
    • B.$x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots$
    • C.$1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots$
    • D.$1 - x + x^2 - x^3 + \cdots$
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    Correct answer: $1 + x + x^2 + x^3 + \cdots$

    The series for $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 + \cdots$, the geometric series with ratio $x$, valid for $|x| < 1$. This is one of the standard Taylor series of common functions. The alternating version $1 - x + x^2 - \cdots$ corresponds instead to $\frac{1}{1 + x}$.

53. Evaluate the improper integral $\int_{1}^{\infty} \frac{1}{x^2}\, dx$.

    • A.Diverges
    • B.1
    • C.2
    • D.$\frac{1}{2}$
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    Correct answer: 1

    The improper integral equals $1$. The antiderivative of $\frac{1}{x^2}$ is $-\frac{1}{x}$; evaluating from $1$ to $b$ gives $-\frac{1}{b} + 1$, and as $b \to \infty$, $-\frac{1}{b}$ approaches $0$, leaving $1$. This contrasts with the integral of $\frac{1}{x}$, which diverges.

54. A particle's position is $s(t) = t^3 - 6t^2 + 9t$. At what time $t > 0$ does the particle change direction for the first time?

    • A.$t = 3$
    • B.$t = 1$
    • C.$t = 2$
    • D.$t = 0$
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    Correct answer: $t = 1$

    The particle first changes direction at $t = 1$. Velocity is $s'(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3)$; the velocity changes sign at $t = 1$, going from positive to negative. The other root $t = 3$ is the second direction change, not the first.

55. What is the value of the integral $\int_{0}^{\pi/2} \sin(x)\cos(x)\, dx$?

    • A.$\frac{1}{2}$
    • B.0
    • C.$\frac{\pi}{4}$
    • D.1
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    Correct answer: $\frac{1}{2}$

    The integral equals $\frac{1}{2}$. Using the substitution $u = \sin(x)$, $du = \cos(x)\, dx$, the integral becomes $\int_{0}^{1} u\, du = \frac{1}{2}$. Alternatively, $\sin(x)\cos(x) = \frac{1}{2}\sin(2x)$, whose integral over this interval also gives $\frac{1}{2}$.

56. Find the local minimum value of $f(x) = x^3 - 3x$ on its domain of all real numbers.

    • A.$-2$
    • B.$-1$
    • C.0
    • D.2
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    Correct answer: $-2$

    The local minimum value is $-2$. Setting $f'(x) = 3x^2 - 3 = 0$ gives critical points $x = 1$ and $x = -1$; the second derivative $f''(x) = 6x$ is positive at $x = 1$, indicating a local minimum, and $f(1) = 1 - 3 = -2$. The point $x = -1$ yields a local maximum value of $2$.

57. What is the arc length of the curve $y = \frac{2}{3} x^{3/2}$ from $x = 0$ to $x = 3$?

    • A.$\frac{14}{3}$
    • B.6
    • C.$\frac{13}{3}$
    • D.$\frac{16}{3}$
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    Correct answer: $\frac{14}{3}$

    The arc length is $\frac{14}{3}$. The formula is $\int \sqrt{1 + (y')^2}\, dx$; here $y' = x^{1/2}$, so $1 + (y')^2 = 1 + x$, and $\int_{0}^{3} \sqrt{1 + x}\, dx = \frac{2}{3}(1 + x)^{3/2}$ evaluated from $0$ to $3$ equals $\frac{2}{3}(8 - 1) = \frac{14}{3}$.

58. Evaluate $\lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^x$.

    • A.3
    • B.$e^3$
    • C.$e$
    • D.$\infty$
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    Correct answer: $e^3$

    The limit equals $e^3$. The standard limit $\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a$; here $a = 3$, so the value is $e^3$. This is the continuous-compounding form of the exponential limit definition.

59. What is the derivative of $f(x) = \arcsin(x)$?

    • A.$-\frac{1}{\sqrt{1 - x^2}}$
    • B.$\frac{1}{1 + x^2}$
    • C.$\frac{1}{\sqrt{x^2 - 1}}$
    • D.$\frac{1}{\sqrt{1 - x^2}}$
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    Correct answer: $\frac{1}{\sqrt{1 - x^2}}$

    The derivative of $\arcsin(x)$ is $\frac{1}{\sqrt{1 - x^2}}$. This is a standard inverse-trigonometric derivative valid on $(-1, 1)$. The expression $\frac{1}{1 + x^2}$ is the derivative of $\arctan(x)$, a common point of confusion.

60. Using implicit differentiation, find $\frac{dy}{dx}$ for the curve $x^2 + xy + y^2 = 7$ at a general point.

    • A.$-\frac{2x + y}{2y}$
    • B.$-\frac{2x + y}{x + 2y}$
    • C.$-\frac{x + 2y}{2x + y}$
    • D.$\frac{2x + y}{x + 2y}$
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    Correct answer: $-\frac{2x + y}{x + 2y}$

    The derivative is $-\frac{2x + y}{x + 2y}$. Differentiating implicitly gives $2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$; collecting $\frac{dy}{dx}$ terms yields $(x + 2y)\frac{dy}{dx} = -(2x + y)$, so $\frac{dy}{dx} = -\frac{2x + y}{x + 2y}$. The product rule on the $xy$ term contributes both $y$ and $x\frac{dy}{dx}$.

61. What is the value of the integral $\int_{0}^{1} \frac{1}{1 + x^2}\, dx$?

    • A.$\ln 2$
    • B.$\frac{\pi}{2}$
    • C.$\frac{\pi}{4}$
    • D.1
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    Correct answer: $\frac{\pi}{4}$

    The integral equals $\frac{\pi}{4}$. The antiderivative of $\frac{1}{1 + x^2}$ is $\arctan(x)$; evaluating from $0$ to $1$ gives $\arctan(1) - \arctan(0) = \frac{\pi}{4} - 0 = \frac{\pi}{4}$. This is a frequently used definite integral connecting calculus to the value of $\pi$.

62. For the function $f(x, y) = x^2 - y^2$, classify the critical point at the origin using the second derivative test.

    • A.Saddle point
    • B.Inconclusive
    • C.Local maximum
    • D.Local minimum
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    Correct answer: Saddle point

    The origin is a saddle point. The second-derivative (Hessian) test computes $D = f_{xx} f_{yy} - (f_{xy})^2 = (2)(-2) - 0 = -4$; a negative discriminant $D$ indicates a saddle point regardless of the signs of the individual second partials. The function increases along the $x$-axis and decreases along the $y$-axis.

63. Evaluate $\lim_{x \to 0^+} x \ln(x)$.

    • A.Does not exist
    • B.0
    • C.1
    • D.$-\infty$
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    Correct answer: 0

    The limit is $0$. Rewriting $x \ln(x)$ as $\frac{\ln(x)}{1/x}$ gives the indeterminate form $\frac{-\infty}{\infty}$, and L'Hopital's rule differentiates to $\frac{1/x}{-1/x^2} = -x$, which approaches $0$. The product of a quantity approaching $0$ and one approaching $-\infty$ resolves to $0$ here.

64. What is the average value of $f(x) = x^2$ on the interval $[0, 3]$?

    • A.4.5
    • B.9
    • C.3
    • D.6
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    Correct answer: 3

    The average value is $3$. The average value of a function on $[a, b]$ is $\frac{1}{b - a} \int_{a}^{b} f(x)\, dx$; here it is $\frac{1}{3} \int_{0}^{3} x^2\, dx = \frac{1}{3}(9) = 3$. The integral itself equals $9$, but the average requires dividing by the interval length.

65. Evaluate the integral over the region bounded by switching the order: $\int_{0}^{1} \int_{y}^{1} e^{x^2}\, dx\, dy$.

    • A.$e - 1$
    • B.$\frac{e - 1}{2}$
    • C.$\frac{e^2 - 1}{2}$
    • D.$\frac{1}{2}$
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    Correct answer: $\frac{e - 1}{2}$

    The value is $\frac{e - 1}{2}$. Because $e^{x^2}$ has no elementary antiderivative, reversing the order of integration to $\int_{0}^{1} \int_{0}^{x} e^{x^2}\, dy\, dx$ gives $\int_{0}^{1} x e^{x^2}\, dx$, which by substitution $u = x^2$ equals $\frac{1}{2}(e - 1)$. Recognizing when to switch the order is essential here.

66. What is the curvature of a circle of radius $5$?

    • A.$\frac{1}{5}$
    • B.$\frac{1}{25}$
    • C.25
    • D.5
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    Correct answer: $\frac{1}{5}$

    The curvature of a circle of radius $5$ is $\frac{1}{5}$. Curvature is defined as the reciprocal of the radius of the osculating circle, so for a circle of radius $r$ the curvature is constant and equals $\frac{1}{r}$. A larger radius corresponds to a gentler bend and thus smaller curvature.

67. Using partial fractions, evaluate $\int \frac{1}{x^2 - 1}\, dx$.

    • A.$\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$
    • B.$\ln |x^2 - 1| + C$
    • C.$\arctan(x) + C$
    • D.$\frac{1}{2} \ln |x^2 - 1| + C$
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    Correct answer: $\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$

    The antiderivative is $\frac{1}{2} \ln \left| \frac{x - 1}{x + 1} \right| + C$. Decomposing $\frac{1}{x^2 - 1}$ into $\frac{1/2}{x - 1} - \frac{1/2}{x + 1}$ and integrating each term gives $\frac{1}{2} \ln|x - 1| - \frac{1}{2} \ln|x + 1|$, which combines into $\frac{1}{2} \ln$ of the ratio. Partial fraction decomposition is the standard technique for rational integrands with factorable denominators.

68. Using Green's theorem, evaluate the counterclockwise line integral $\oint -y\, dx + x\, dy$ around the circle $x^2 + y^2 = 4$.

    • A.$16\pi$
    • B.$4\pi$
    • C.$2\pi$
    • D.$8\pi$
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    Correct answer: $8\pi$

    The line integral equals $8\pi$. Green's theorem converts the boundary integral of $P\, dx + Q\, dy$ into the double integral of $\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}$ over the enclosed region; with $P = -y$ and $Q = x$, the integrand is $1 - (-1) = 2$, so the value is $2$ times the enclosed area. The disk of radius $2$ has area $4\pi$, giving $2 \times 4\pi = 8\pi$. Forgetting the factor of $2$ yields the tempting $4\pi$.

69. Apply the limit comparison test to determine whether the series $\sum_{n=1}^{\infty} \frac{2n + 1}{n^3 + n}$ converges.

    • A.The test is inconclusive
    • B.Converges, comparing with $\sum \frac{1}{n^2}$
    • C.Converges, comparing with $\sum \frac{1}{n}$
    • D.Diverges, comparing with $\sum \frac{1}{n}$
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    Correct answer: Converges, comparing with $\sum \frac{1}{n^2}$

    The series converges by comparison with $\sum \frac{1}{n^2}$. The limit comparison test takes the ratio of the term $\frac{2n + 1}{n^3 + n}$ to $\frac{1}{n^2}$, which simplifies to $\frac{2n^3 + n^2}{n^3 + n}$ and approaches the finite positive limit $2$. Because the comparison series $\sum \frac{1}{n^2}$ converges as a p-series with $p = 2$, the original series converges as well.

70. Determine whether the geometric series $\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n$ converges, and if so to what value.

    • A.Converges to $\frac{2}{3}$
    • B.Diverges
    • C.Converges to $3$
    • D.Converges to $\frac{3}{2}$
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    Correct answer: Converges to $3$

    The geometric series converges to $3$. A geometric series with first term $a$ and common ratio $r$ converges when $|r| < 1$, and its sum is $\frac{a}{1 - r}$. Here $a = 1$ (the $n = 0$ term) and $r = \frac{2}{3}$, so the sum is $\frac{1}{1 - 2/3} = \frac{1}{1/3} = 3$. The value $\frac{3}{2}$ mistakenly uses a ratio of $\frac{1}{3}$ instead of $\frac{2}{3}$.

71. Find the volume of the solid generated by revolving the region bounded by $y = \sqrt{x}$, $y = 0$, and $x = 4$ about the x-axis.

    • A.$8\pi$
    • B.$4\pi$
    • C.$16\pi$
    • D.$\dfrac{16}{3}\pi$
    Show answerHide answer

    Correct answer: $8\pi$

    Correct answer: $8\pi$. Using the disk method, the volume is $V = \pi \int_{0}^{4} (\sqrt{x})^2\, dx = \pi \int_{0}^{4} x\, dx = \pi \left[ \tfrac{x^2}{2} \right]_0^4 = \pi \cdot 8 = 8\pi$.

72. Using the quotient rule, what is the derivative of $f(x) = \dfrac{x^2 + 1}{x - 1}$?

    • A.$\dfrac{2x}{1}$
    • B.$\dfrac{x^2 - 2x - 1}{(x-1)^2}$
    • C.$\dfrac{x^2 + 2x - 1}{(x-1)^2}$
    • D.$\dfrac{2x(x-1) + (x^2+1)}{(x-1)^2}$
    Show answerHide answer

    Correct answer: $\dfrac{x^2 - 2x - 1}{(x-1)^2}$

    Correct answer: $\dfrac{x^2 - 2x - 1}{(x-1)^2}$. The quotient rule gives $f'(x) = \dfrac{(2x)(x-1) - (x^2+1)(1)}{(x-1)^2} = \dfrac{2x^2 - 2x - x^2 - 1}{(x-1)^2} = \dfrac{x^2 - 2x - 1}{(x-1)^2}$.

73. At what x-value does the curve $y = x^3 - 6x^2 + 5$ have an inflection point?

    • A.$x = 4$
    • B.$x = 0$
    • C.$x = 2$
    • D.$x = 3$
    Show answerHide answer

    Correct answer: $x = 2$

    Correct answer: $x = 2$. An inflection point occurs where the second derivative changes sign. Here $y' = 3x^2 - 12x$ and $y'' = 6x - 12$; setting $y'' = 0$ gives $6x - 12 = 0$, so $x = 2$, and the concavity changes there.

74. For which values of $p$ does the p-series $\sum_{n=1}^{\infty} \dfrac{1}{n^p}$ converge?

    • A.$p \geq 1$
    • B.$p > 1$
    • C.$p < 1$
    • D.All real $p$
    Show answerHide answer

    Correct answer: $p > 1$

    Correct answer: $p > 1$. The p-series $\sum 1/n^p$ converges precisely when $p > 1$ and diverges when $p \leq 1$; for example $p = 1$ gives the divergent harmonic series.