Question 1

Find the absolute maximum and minimum values of $$f(t) = t + 2\cos t on $$the interval $$[0, 2\pi]. $$Which of the following is correct?

Accepted Answer

Absolute maximum at $$t = 2\pi$$, minimum at $$t = \pi$$

Question 2

Given $$f(x) = \sqrt{x+1} on$$ $$[-1, 3]$$, which of the following statements is TRUE regarding the Mean Value Theorem (MVT)?

Accepted Answer

The function is continuous on $$[-1, 3]$$ and differentiable on $$(-1, 3)$$, so MVT applies.

Question 3

For $$f(x) = \sqrt{x+1} on$$ $$[-1, 3]$$, find all $$c in$$ $$(-1, 3)$$ that satisfy the conclusion of the Mean Value Theorem.

Accepted Answer

$$c = 1$$

Question 4

Find the local maximum and minimum values, if they exist, of $$h(x) = x^2$ \ln x.$$

Accepted Answer

Local minimum at $$x = e$$^{-1/2}$$, local maximum at x = 1$

Question 5

Given $$g(x) = x^4$ - 2x^2$ + 3$$, which of the following is a critical number?

Accepted Answer

$$x = 1$$

Question 6

For $$g(x) = x^4$ - 2x^2$ + 3$$, on which open interval is the function increasing?

Accepted Answer

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

Question 7

Evaluate $$\displaystyle \lim_{t 	o 0} t \csc t.$$

Accepted Answer

$$0$$

Question 8

Evaluate $$\displaystyle \lim_{x 	o \infty} e$$^{\frac{1}$${x} - 1}.$$

Accepted Answer

$$e^{-1}$$

Question 9

Evaluate $$\displaystyle \lim_{x 	o -\infty} x^2$ e$$^{2x}$$.

Accepted Answer

$$0$$

Question 10

Evaluate $$\displaystyle \lim_{x 	o 0} x^x.$$

Accepted Answer

$$1$$

Question 11

A jet flies at $$600 ft/sec at a $$constant elevation of $$900 ft$$, passing directly over a spectator. How quickly is the angle of elevation $$	heta$$ changing $$2$$ seconds later?

Accepted Answer

Question 12

Use linear approximation to estimate $$\sqrt{35}. $$Which of the following is closest to the estimate?

Accepted Answer

$$5.92$$

Question 13

Use linear approximation to estimate $$\pi^2$ using x = 3$ as the base point.

Accepted Answer

$$9.87$$

Question 14

Suppose you have $$102$$ meters of fencing to make two side-by-side rectangular enclosures (sharing one side). What is the maximum area you can enclose?

Accepted Answer

$$676$$ $$m^2$

Question 15

For $$g(x) = x^4$ - 2x^2$ + 3$$, at which $$x-$$value does the function have an inflection point?

Accepted Answer

$$x = 0$$

Question 16

For $$g(x) = x^4$ - 2x^2$ + 3$$, on which interval is the function concave up?

Accepted Answer

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

Question 17

Which of the following best describes the behavior of $$h(x) = x^2$ \ln x as$$ $$x 	o 0^+$$?

Accepted Answer

$$h(x) 	o -0$$

Question 18

A chemical reaction's rate is modeled by $$R(t) = t + 2\cos t$$ for $$t in$$ $$[0, 2\pi]. At $$what time is the reaction rate minimized?

Accepted Answer

$$t = \pi$$

Calculus Test 3 Review – Step-by-Step Guidance

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