Question 1

Given the function $$f(x) = 4x^2$ - 12x + 9 on $$the interval $$[0,2]$$, what is the absolute minimum value of $$f on $$this interval?

Accepted Answer

$$f(\frac{3}{2}) = 0$$

Question 2

For the function $$g(x) = x \ln(x)$$ defined on $$(0, \infty)$$, at which $$x$$ does $$g$$ have a critical point?

Accepted Answer

Question 3

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, which equation must have a solution $$c in$$ $$(a, b)$$?

Accepted Answer

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Question 4

Given the graph of $$f'(x)$$, at which $$x-$$values does $$f(x)$$ have inflection points?

Accepted Answer

Where $$f'(x)$$ has a local maximum or minimum

Question 5

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

Accepted Answer

$$3$$

Question 6

For the function $$f(x) = 4x^2$ - 4x + 1 on$$ $$(-\infty, \infty)$$, what is the concavity of $$f$$?

Accepted Answer

Concave up everywhere

Question 7

Find the critical point of $$f(x) = 4x^2$ - 4x + 1.$$

Accepted Answer

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

Question 8

Suppose $$xy = 24$$ and you want to minimize $$3x + y$$ for positive $$x$$ and $$y. $$What is the minimum value of $$x$$?

Accepted Answer

$$x = \sqrt{8}$$

Question 9

Given $$f(x) = \frac{2}{2 + x}$$, what is the linear approximation $$L(x) at$$ $$x = 1$$?

Accepted Answer

$$L(x) = \frac{2}{3} - \frac{2}{9}(x - 1)$$

Question 10

Evaluate $$\lim_{x 	o 0^+} \sqrt{x} \ln(x).$$

Accepted Answer

$$0$$

Question 11

If $$f(x) = x + \frac{1}{x} on$$ $$[1,3]$$, which $$c in$$ $$(1,3) is $$guaranteed by the Mean Value Theorem?

Accepted Answer

$$c = \sqrt{3}$$

Question 12

Evaluate $$\lim_{x 	o 0^+} x^7$ e$$^{7x}$$.

Accepted Answer

$$0$$

Question 13

When using the linear approximation of $$\ln(x)$$ near $$x = e$$, are the approximations overestimates or underestimates of the true values?

Accepted Answer

Overestimates

Question 14

For $$g(x) = x \ln(x)$$, on which interval is $$g$$ increasing?

Accepted Answer

$$(e^{-1}$$, \infty)$$

Question 15

Given $$f(x) = 4x^2$ - 12x + 9 on$$ $$[0,2]$$, what is the absolute maximum value of $$f$$?

Accepted Answer

$$f(0) = 9$$

Question 16

Evaluate $$\lim_{x 	o \frac{\pi}{2}} \frac{\sin(x)}{x}.$$

Accepted Answer

$$\frac{2}{\pi}$$

Question 17

Suppose $$f'(x) is $$positive on $$(-\infty, -2). $$What can you say about $$f(x) on $$this interval?

Accepted Answer

$$f(x) is $$increasing

Question 18

If $$f''(x) \u003e 0$$ for all $$x$$, what can you conclude about the graph of $$f$$?

Accepted Answer

It is concave up everywhere

Question 19

For the function $$f(x) = 4x^2$ - 4x + 1$$, use the second derivative test to classify the critical point at $$x = \frac{1}{2}.$$

Accepted Answer

Local minimum

Calculus Exam Study Guide: Critical Points, Extrema, Concavity, Limits, and Approximations

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