Question 1

If $$f'(c) = 0$$, then which of the following is true about $$f at$$ $$x = c$$?

Accepted Answer

$$x = c is a $$candidate for a local extreme point.

Question 2

The critical points of the function $$f(x) = x^3$ - 3x^2$ + 2$$ are:

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 3

A function $$f$$ has the property that $$f'(c) = 0$$, $$f''(c) \u003e 0$$, and $$f is $$defined at $$c. $$What can you conclude?

Accepted Answer

$$f$$ has a local minimum at $$x = c.$$

Question 4

Suppose the graph of a function $$f$$ satisfies $$f'(x) \u003e 0$$ and $$f''(x) \u003c 0 on an $$interval. Which statement is true?

Accepted Answer

$$f is $$increasing at a decreasing rate.

Question 5

The rational function $$f(x) = \frac{x}{x-1}$$ has:

Accepted Answer

A zero, a vertical asymptote, and a horizontal asymptote.

Question 6

Suppose you want to maximize the area $$A = xy of a $$rectangle, subject to the condition $$2x + y = 100. $$What are the primary and secondary equations?

Accepted Answer

Primary equation: $$A = xy$$; Secondary equation: $$2x + y = 100$$

Question 7

The product of a pair of real numbers that have a sum of 20 is $$x(20-x). $$This product has a maximum when:

Accepted Answer

$$x = 10$$

Question 8

The linear approximation to $$f(x) = \sqrt{x} at$$ $$x = 9 is$$:

Accepted Answer

$$L(x) = 3 + \frac{1}{6}(x-9)$$

Question 9

Using the linear approximation to $$f(x) = \sqrt{x} at$$ $$x = 9$$, approximate $$\sqrt{9.2}.$$

Accepted Answer

3.033

Question 10

If $$y = x^3$, then the differential dy$ is:

Accepted Answer

$$3x^2 dx$

Question 11

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

Accepted Answer

1

Question 12

Evaluate $$\lim_{x 	o \infty} \frac{2x^2$ + 3}{x^2$ - 1}.$$

Accepted Answer

2

Question 13

A dairy farmer wants to build two identical adjacent rectangular pastures next to a barn, using the barn as one side. If the total area is $$15,000 m$^2$$, what dimensions minimize the amount of fencing used?

Accepted Answer

Width $$x = 50 m$$, Length $$y = 150 m$$

Question 14

Suppose $$f(x) = x^3$ - 3x^2$ + 2. On $$which intervals is $$f$$ increasing?

Accepted Answer

$$(-\infty, 0)$$ and $$(2, \infty)$$

Question 15

Find the point of inflection for $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 1$$

Question 16

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}$$ using L'Hôpital's Rule.

Accepted Answer

1

Question 17

Suppose $$f(x) = \sin x on$$ $$[0, \pi]. $$What is the absolute maximum value?

Accepted Answer

1

Question 18

Suppose $$f(x) = \sin x on$$ $$[0, \pi]. $$What is the absolute minimum value?

Accepted Answer

0

Question 19

If $$f''(x) \u003e 0 on an $$interval, what can you say about $$f on $$that interval?

Accepted Answer

$$f is $$concave up.

Calculus I Practice Test 3 – Step-by-Step Study Guidance

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