Question 1

Let $$f(x) = \ln(x) - x. $$What is $$\frac{d}{dx}f(x)$$?

Accepted Answer

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

Question 2

Suppose $$f(x) = x^2$ + 2x + 4 on $$the interval $$[-3,3]. $$What is the maximum value attained by $$f(x) on $$this interval?

Accepted Answer

$$25$$

Question 3

Let $$f(x) = \log_3$(x^2$$)$$. What is f$$'(x)$?

Accepted Answer

$$\frac{2}{x \ln(3)}$$

Question 4

A particle moves along a line with position function $$s(t). If$$ $$s'(t) \u003e 0$$ and $$s''(t) \u003c 0 on an $$interval, what can you conclude about the particle's motion?

Accepted Answer

The particle is moving right and slowing down.

Question 5

Suppose $$f'(x) = x^2$ + 1. $$Which of the following is true about $$f(x)$$?

Accepted Answer

$$f(x) is $$increasing everywhere.

Question 6

The sum of two positive numbers, $$a$$ and $$b$$, is $$36. $$What function would you use to minimize the sum of their squares?

Accepted Answer

$$f(a) = a^2$ + (36 - a)^2$

Question 7

Compute $$\lim_{x 	o \infty} x \sin\left(\frac{2\pi}{x}ight).$$

Accepted Answer

$$2\pi$$

Question 8

Suppose $$f is $$continuous and differentiable on $$[-1,1]. $$Which condition guarantees that $$f'(c) = 0$$ for some $$c$$ between $$-1$$ and $$1$$?

Accepted Answer

$$f(-1) = f(1)$$

Question 9

Let $$y = 	an$$^{-1}$$\left(\frac{a}{x}ight)$$, where $$a is a $$constant. Find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 10

If $$f(x) is $$concave up on an interval, and you use the linearization at $$x=0 to $$estimate $$f(1)$$, will the result be an overestimate or underestimate?

Accepted Answer

Underestimate, because the tangent line lies below the curve.

Question 11

Given the graph of $$f'(x)$$, on what interval(s) is $$f(x)$$ decreasing?

Accepted Answer

$$(-\infty, -1)$$ and $$(1,3)$$

Question 12

At what $$x-$$values does $$f(x)$$ have local minima if $$f'(x)$$ changes from negative to positive?

Accepted Answer

$$x = 1$$

Question 13

On what interval(s) is $$f(x)$$ concave down if $$f''(x) \u003c 0$$?

Accepted Answer

$$(-\infty, -1)$$ and $$(3, \infty)$$

Question 14

A chemical reaction's rate is modeled by $$r(t) = t^2$ e$$^{-t}$$. At what time t$ is the reaction rate maximized?

Accepted Answer

$$t = 2$$

Question 15

A hospital monitors the temperature of a patient modeled by $$T(t) = 98.6 + 2e$$^{-0.5t}$$, where t$ is in hours. What is the patient's temperature after 4 hours?

Accepted Answer

$$99.0$$

Question 16

A company wants to build a rectangular enclosure with a fixed perimeter of $$100$$ meters. What dimensions maximize the area?

Accepted Answer

$$25 m by$$ $$25 m$$

Question 17

Evaluate $$\$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 18

Which of the following is the correct derivative of $$y = \arcsin(x)$$?

Accepted Answer

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

Question 19

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. At $$what time does it reach its highest point?

Accepted Answer

$$t = 2.04 s$$

Calculus I: Differentiation, Applications, and Optimization Study Guide

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