$$f'(2)g(2) + f(2)g'(2) = 1 \times 2 + 3 \times 4 = 2 + 12 = 14$$

Question 1

Given the curve defined by $$x^2 + xy$$ + $$y^2 = 27$, what is the expression for the slope of the curve at any point (x$$, y)$$?

Accepted Answer

$$\frac{-2x - y}{x + 2y}$$

Question 2

For the curve $$x^2 + xy$$ + $$y^2 = 27$, at which points are the tangent lines to the curve vertical?

Accepted Answer

$$(3, 6)$$ and $$(-3, -6)$$

Question 3

Consider the function $$y = 3x^3$ + 3x^2$. What is $$\frac{dy}{dx}$$ at x = 0$?

Accepted Answer

$$0$$

Question 4

Find the equation of the tangent line to $$y = \frac{1}{5}(x-2) at $$the point $$(2, 0).$$

Accepted Answer

$$y = \frac{1}{5}(x-2)$$

Question 5

Given $$f(x) = \sin(e^x)$$, what is $$f'(x)$$?

Accepted Answer

$$\cos(e^x) \cdot e^x$$

Question 6

If $$g(x) = \ln(x^2$ + 2x)$$, what is $$g'(x)$$?

Accepted Answer

$$\frac{2x + 2}{x^2$ + 2x}$$

Question 7

Find $$\frac{d}{dx} \arctan(6x).$$

Accepted Answer

$$\frac{6}{1 + 36x^2$}$$

Question 8

Given $$h(x) = f(x)g(x)$$, and $$f(2) = 3$$, $$g(2) = 2$$, $$f'(2) = 1$$, $$g'(2) = 4$$, what is $$h'(2)$$?

Accepted Answer

$$f'(2)g(2) + f(2)g'(2) = 1 	imes 2 + 3 	imes 4 = 2 + 12 = 14$$

Question 9

Evaluate $$\lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 10

Which of the following functions is NOT differentiable at $$x = 1$$?

Accepted Answer

$$f(x) = |x-1|$$

Question 11

Given $$y = x^2$ + 2x + 1$$, find the rate of change of the slope at $$x = 0.$$

Accepted Answer

$$2$$

Question 12

If $$f(x) = e$$^{x^2}$$, what is f$$'(x)$$?

Accepted Answer

$$2x e$$^{x^2}$$

Question 13

Let $$y = \ln(x^2$ + 2x)$$, what is $$\frac{dy}{dx} at$$ $$x = 1$$?

Accepted Answer

$$1$$

Question 14

Find the derivative of $$f(x) = \log_8$(3x-7).$$

Accepted Answer

$$\frac{3}{(3x-7) \ln 8}$$

Question 15

Given $$f(x) = x^2$, g(x) = 2x$, find $$\frac{d}{dx}[g(2x-x) - (2x-x)]$$ at x = 1$.

Accepted Answer

$$2$$

Question 16

If $$f(x) = \sin x$$, what is $$\frac{d}{dx} f$$^{-1}$$(x) at$$ $$x = \frac{1}{2}$$?

Accepted Answer

$$\frac{1}{\cos(f$$^{-1}$$(x))}$$

Question 17

Find the $$x-$$coordinate(s) where the curve $$y = x^4$ - 12x^2$ + 10$$ has a horizontal tangent.

Accepted Answer

$$x = -\sqrt{6}, \sqrt{6}$$

Question 18

Evaluate $$\lim_{x 	o \infty} \frac{6x^2$ + 2x}{3x^2$ - x}.$$

Accepted Answer

$$2$$

Question 19

Which of the following is a removable discontinuity?

Accepted Answer

$$f(x) = \frac{x^2$ - 1}{x - 1} at$$ $$x = 1$$

Calculus I: Derivatives, Tangents, and Continuity – Unit 3 Review Study Guide

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