Q: The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 ft/min. When the top of the ladder is 17 ft from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

$$\frac{3 \times 17}{\sqrt{25^2$ - 17^2$}} ft/$$min

Q: Use logarithmic differentiation to find the derivative of $$y = (\cos x)^x.$$

$$(\cos x)^x [\ln(\cos x) - x \tan x]$$

Question 1

Given the graph of $$f(x)$$ shown, what is $$\lim\limits_{x 	o 2^-} f(x)$$?

Accepted Answer

$$-1$$

Question 2

Suppose $$f(x) is $$continuous for all real numbers and $$f(x) = \frac{x^2$ - 4}{x - 2}$$ when $$x 
eq 2$$, and $$f(2) = k. $$What value of $$k$$ makes $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

$$4$$

Question 3

Let $$f(x) = \sqrt{x - 2}$$ and $$y = (x^2$ + 10)x. $$What is $$\frac{dy}{dx} at $$the point where $$x = 17$$?

Accepted Answer

$$867$$

Question 4

A particle moves along a line with position $$s(t) = 2t^3$ - 21t^2$ + 108t + 53$$ for $$t \geq 0. $$For what value of $$t is $$the particle at rest?

Accepted Answer

$$t = 3$$

Question 5

Given $$y = 21x^2$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$42x$$

Question 6

The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 ft/min. When the top of the ladder is 17 ft from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

Accepted Answer

$$\frac{3 	imes 17}{\sqrt{25^2$ - 17^2$}} ft/$$min

Question 7

If $$f(x) = 	an(2x)$$, then $$f'(x) = $$

Accepted Answer

$$2 \sec^2$(2x)$$

Question 8

Use logarithmic differentiation to find the derivative of $$y = (\cos x)^x.$$

Accepted Answer

$$(\cos x)^x [\ln(\cos x) - x 	an x]$$

Question 9

The radius of a circle is increasing. At a certain instant, the rate of increase in the area of the circle is numerically equal to twice the rate of increase in its circumference. What is the radius of the circle at that instant?

Accepted Answer

$$r = 2$$

Question 10

Let $$f be a $$continuous function on $$[2,5]. If$$ $$f(2) = -5$$, $$f(5) = 5$$, and $$f(9) = -5$$, which of the following must be true?
I. $$f$$ has at least 2 zeros
II. For some $$c$$, $$2 \u003c c \u003c 5$$, $$f(c) = 3$$

Accepted Answer

Both I and II

Question 11

Find $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 12

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 13

If $$f(x) = \ln(x)$$, what is $$\lim\limits_{x 	o 0^+} f(x)$$?

Accepted Answer

$$-\infty$$

Question 14

Given $$y = \log_2$(8x)$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$\frac{1}{x \ln 2}$$

Question 15

Find $$\lim\limits_{x 	o 3} \frac{\sqrt{x+1} - 2}{x-3}.$$

Accepted Answer

$$\frac{1}{4}$$

Question 16

If $$f(x) = x^3$ - 13$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2$$

Question 17

Given $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

$$-\frac{1}{x^2$}$$

Question 18

If $$f(x) = \sec x$$, what is $$f'(x)$$?

Accepted Answer

$$\sec x 	an x$$

Question 19

Given $$f(x) = x^2$ + 7x + 11$$, what is $$f'(x)$$?

Accepted Answer

$$2x + 7$$

Question 20

A position function is $$s(t) = 30t - 5t^2$ for 0$$ \leq t \leq 8$$. When is the object stationary?

Accepted Answer

$$t = 3$$

Calculus I: Limits, Continuity, and Derivatives – Study Guide

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