Praxis Mathematics 5165 Domain 2: Functions and Calculus

Given the function \( f(x) = x^3 - 6x^2 + 11x - 6 \), for which values of \( x \) does \( f(x) = 0 \)?

What is the derivative of the function \( g(x) = \frac{x^2 - 1}{x^2 + 1} \)?

What is the limit of \( \frac{\sin(x)}{x} \) as \( x \) approaches 0?

Calculate the integral \( \int (3x^2 - 2x + 1) \, dx \).

What is the area under the curve of \( f(x) = 4 - x^2 \) from \( x = -2 \) to \( x = 2 \)?

What is the derivative of \( f(x) = \ln(\sqrt{x^2 + 1}) \)?

Evaluate the limit \( \lim_{x \to \infty} \frac{3x^3 + 2x^2 - x}{4x^3 - x^2 + 5} \).

If \( h(x) = e^{2x} \), what is the second derivative of \( h \) at \( x = 0 \)?

Find the critical points of the function \( f(x) = x^4 - 4x^2 \).

What is the convergence interval of the power series \( \sum_{n=1}^{\infty} \frac{(x-1)^n}{n2^n} \)?

Calculate the volume of the solid formed by revolving the area between \( y = x^2 \) and \( y = x \) around the x-axis.

Find the derivative of the function \( f(x) = x^2e^{3x} \).

What is the limit as \( x \) approaches infinity of \( \frac{e^x}{x^2} \)?

Calculate the area between the curves \( y = \sin x \) and \( y = \cos x \) from \( x = 0 \) to \( x = \frac{\pi}{4} \).

If \( f(x) = \tan^{-1}(x) \), what is \( f''(0) \)?

Evaluate the integral \( \int_0^1 \ln(1 + x) \, dx \).

What is the Maclaurin series expansion of \( e^{x^2} \)?

Find the Taylor series for \( \cos(x) \) centered at \( \pi \).

Determine the radius of convergence for the series \( \sum_{n=1}^{\infty} \frac{n!x^n}{n^n} \).

If \( f(x) = \int_{0}^{x} t^2 e^{t^3} dt \), what is \( f'(x) \)?