GRE Math Domain 1: Calculus Welcome to your GRE Math Domain 1: Calculus 1. GRE Math: Calculus 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. \( 2e^{2x} \cos(x) - e^{2x} \sin(x) \) None 2. GRE Math: Calculus 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. \( \frac{x}{\ln(x)} + C \) None 3. GRE Math: Calculus 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 None 4. GRE Math: Calculus 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 None 5. GRE Math: Calculus If \( f(x) = \ln(x) \), what is \( f''(e) \)? A. \( \frac{1}{e} \) B. \( -\frac{1}{e} \) C. \( \frac{1}{e^2} \) D. \( -\frac{1}{e^2} \) None 6. GRE Math: Calculus What is the volume of the solid obtained by rotating the region bounded by \( y = x^3 \), \( y = 0 \), and \( x = 2 \) around the y-axis? A. \( 16\pi \) B. \( 32\pi \) C. \( 64\pi \) D. \( 128\pi \) None 7. GRE Math: Calculus 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. \( \frac{1}{2}x^{-\frac{1}{2}} + 2x^\frac{3}{2} \) None 8. GRE Math: Calculus Find the equation of the tangent line to the curve \( y = x^3 - 3x \) at \( x = 2 \). A. \( y = 9x - 14 \) B. \( y = 9x + 14 \) C. \( y = 12x - 20 \) D. \( y = 12x + 20 \) None 9. GRE Math: Calculus Evaluate the definite integral \( \int_{0}^{2} (3x^2 - 2x) dx \). A. 4 B. 6 C. 8 D. 10 None 10. GRE Math: Calculus 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. \( t^4 + 4t^3 \ln(t) \) None 11. GRE Math: Calculus 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. \( x^3 + x^2 + x + C \) None 12. GRE Math: Calculus Determine the limit: \( \lim_{x \to \infty} \frac{4x^3 - x}{2x^3 + 5} \). A. 0 B. 2 C. \( \frac{4}{2} \) D. \( \infty \) None 13. GRE Math: Calculus What is the second derivative of \( f(x) = \cos(x^2) \)? A. \( -2\sin(x^2) \) B. \( -4x^2 \cos(x^2) \) C. \( -2x \sin(x^2) \) D. \( -4x \sin(x^2) \) None 14. GRE Math: Calculus 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. \( \frac{e^2}{2} \) None 15. GRE Math: Calculus Calculate the derivative of \( h(x) = \frac{x^2 + 1}{x^2 - 1} \). A. \( \frac{-2}{(x^2 - 1)^2} \) B. \( \frac{2x^3 - 2x}{(x^2 - 1)^2} \) C. \( \frac{4x}{(x^2 - 1)^2} \) D. \( \frac{2x^3 + 2x}{(x^2 - 1)^2} \) None 16. GRE Math: Calculus Evaluate: \( \lim_{x \to 0} \frac{\sin(3x)}{x} \). A. 0 B. 1 C. 3 D. Undefined None 17. GRE Math: Calculus Find the volume of the solid formed by revolving the region bounded by \( y = x^2 \) and \( y = 0 \) around the x-axis between \( x = 0 \) and \( x = 1 \). A. \( \frac{\pi}{3} \) B. \( \frac{\pi}{2} \) C. \( \pi \) D. \( 2\pi \) None 18. GRE Math: Calculus 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. \( x^5 \cos(x^2) \) None 19. GRE Math: Calculus 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. \( \frac{2x + 1}{(x^2 + x + 1)^2} \) None 20. GRE Math: Calculus Evaluate the limit: \( \lim_{x \to 0} \frac{e^{x} - 1}{x} \). A. 0 B. 1 C. \( e \) D. Undefined None 21. GRE Math: Calculus 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. \( \frac{1}{2}\cos(x^2) + C \) None 22. GRE Math: Calculus Compute the area between the curves \( y = x^2 \) and \( y = x \) for \( 0 \leq x \leq 1 \). A. \( \frac{1}{6} \) B. \( \frac{1}{3} \) C. \( \frac{1}{2} \) D. \( \frac{2}{3} \) None 23. GRE Math: Calculus 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 None 24. GRE Math: Calculus If \( f(x) = x^3 - 3x + 2 \), find \( f^{-1}(1) \). A. 0 B. 1 C. 2 D. 3 None 25. GRE Math: Calculus 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 None 26. GRE Math: Calculus 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. \( \arcsec x + C \) D. \( \ln |x^2 - \sqrt{x^2 - 1}| + C \) None 27. GRE Math: Calculus 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. \( \frac{1}{1+y^2} \) None 28. GRE Math: Calculus Compute the limit: \( \lim_{x \to \infty} \frac{\sqrt{x^2 + x} - x}{x} \). A. 0 B. \( \frac{1}{2} \) C. 1 D. Undefined None 29. GRE Math: Calculus 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. \( t^4 + 4t^3 \ln(t) \) None 30. GRE Math: Calculus 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. \( \arcsec x + C \) D. \( \ln |x^2 - \sqrt{x^2 - 1}| + C \) None 1 out of 30 Time is Up! 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