Q: Find the derivative of $$f(x) = \tan^2$(2x) + \cos(2x - \sin x^2$).$$

$$f'(x) = 2\tan(2x)\sec^2$(2x) - 2\sin(2x - \sin x^2$)(1 - 2x\cos x^2$)$$

Question 1

Solve for $$x in $$the equation $$\ln(x - 2) = 3 \ln 2 + \ln(x).$$

Accepted Answer

$$x = 10$$

Question 2

Let $$f(x) = \sqrt{x}$$ and $$g(x) = 2$$^{x^2}$$. What is the domain of the composite function g$$ \circ f$$?

Accepted Answer

$$x \geq 0$$

Question 3

Find the inverse function $$f^{-1}(x$$)$$ for f(x$$) = \ln(8 - x)$$ and state its domain.

Accepted Answer

$$f^{-1}(x) = 8 - e^x$$,\ x \u003c \ln 8$$

Question 4

Evaluate $$\displaystyle \lim_{x 	o 2} \frac{13 - 2x}{x^2$ - 6} or $$explain why it does not exist.

Accepted Answer

$$-1$$

Question 5

Find all vertical asymptotes of the function $$y = \frac{(x^2$ - 9)\sqrt{x^2$ + 9}}{x^2$ - 9}.$$

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 6

Consider the piecewise function $$f(x) = \begin{cases} x+1, & x \u003c 1 \ ax^2$ + b, & 1 \leq x \u003c 2 \ x^2$ + 3, & x \geq 2 \end{cases}. $$For which values of $$a$$ and $$b is$$ $$f(x)$$ continuous everywhere?

Accepted Answer

$$a = 1,\ b = 2$$

Question 7

Find the derivative of $$f(x) = \frac{x^2$ + 2x - 10}{x^2$}.$$

Accepted Answer

$$f'(x) = \frac{2x^2$ - 2x + 20}{x^3$}$$

Question 8

Find the derivative of $$f(x) = (x^2$ + 2x)^e + e^2$.

Accepted Answer

$$f'(x) = e(x^2$ + 2x$$)^{e-1}$$(2x + 2)$$

Question 9

Find the derivative of $$f(x) = 	an^2$(2x) + \cos(2x - \sin x^2$).$$

Accepted Answer

$$f'(x) = 2	an(2x)\sec^2$(2x) - 2\sin(2x - \sin x^2$)(1 - 2x\cos x^2$)$$

Question 10

Find the second derivative of $$f(x) = \frac{\sin(bx)}{1 + \cos(bx)} at$$ $$x = 0$$, where $$b is a $$parameter.

Accepted Answer

$$f''(0) = \frac{b^2$}{2}$$

Question 11

Given $$f(x) = \frac{1}{\sqrt{1 - x^2$}}$$, use the definition of the derivative as a limit to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$f'(x) = \frac{x}{(1 - x^2$)^{3/2}$$}$

Question 12

The tangent line to the curve $$y = f(x) at $$the point $$(2,1) is $$given by which of the following equations if $$f'(2) = -4$$?

Accepted Answer

$$y = -4x + 9$$

Question 13

Assume $$h(x) = f(g(x))$$ where both $$f$$ and $$g$$ are differentiable for all real $$x. $$Given $$g(-1) = 2$$, $$g'(-1) = 3$$, and $$f'(2) = -4$$, what is $$h'(-1)$$?

Accepted Answer

$$-12$$

Question 14

Which of the following best describes the range of the function $$f(x) = \ln(8 - x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 15

If $$f(x) = \sqrt{x}$$ and $$g(x) = 2$$^{x^2}$$, what is the range of the composite function g$$ \circ f$$?

Accepted Answer

$$[1, \infty)$$

Question 16

Which of the following statements about the function $$f(x) = \frac{1}{\sqrt{1 - x^2$}} is $$true?

Accepted Answer

The function is defined for $$|x| \u003c 1.$$

Question 17

Which of the following is the correct limit $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$1$$

Question 18

For the function $$f(x) = \frac{x^2$ + 2x - 10}{x^2$}$$, which of the following statements is true?

Accepted Answer

There is a vertical asymptote at $$x = 0.$$

Calculus I Midterm Study Guide: Functions, Limits, Derivatives, and Continuity

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