Question 1

Given $$f(x) = \sqrt{x}$$, use the linear approximation at $$a=1 to $$estimate $$\sqrt{1.1}. $$Which of the following is the correct linear approximation?

Accepted Answer

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

Question 2

What is the value of the linear approximation $$L(x)$$ for $$f(x) = \sqrt{x} at$$ $$x = 1.1$$ using $$a = 1$$?

Accepted Answer

$$L(1.1) = 1.05$$

Question 3

For $$f(x) = \sqrt[4]{x} at$$ $$a = 81$$, what is the linear approximation $$L(x)$$?

Accepted Answer

$$L(x) = 3 + \frac{1}{108}(x-81)$$

Question 4

Find the critical numbers of $$f(x) = x^2$ \ln x. $$Which of the following is correct?

Accepted Answer

$$x = e$$^{-1/2}$$

Question 5

For $$f(x) = x$$^{3/5}$$(4-x)$$, what is the critical number found by setting $$f'(x) = 0$$?

Accepted Answer

$$x = \frac{12}{8}$$

Question 6

Given $$f(x) = x^2$ e$$^{-3x}$$, what are the critical numbers?

Accepted Answer

$$x = 0$$ and $$x = \frac{2}{3}$$

Question 7

Which of the following steps is NOT part of finding the absolute maximum and minimum values of a continuous function $$f on$$ $$[a, b]$$?

Accepted Answer

Set $$f(x) = 0$$ and solve for $$x$$ only

Question 8

For $$f(x) = x^3$ - 3x^2$ + 1 on$$ $$[0,4]$$, what is the absolute maximum value?

Accepted Answer

$$f(4) = 17$$

Question 9

For $$f(x) = x^3$ - 3x^2$ + 1 on$$ $$[0,4]$$, what is the absolute minimum value?

Accepted Answer

$$f(2) = -3$$

Question 10

Which of the following best describes the linear approximation $$L(x) of a $$function $$f at$$ $$x=a$$?

Accepted Answer

$$L(x) \approx f'(a)(x-a) + f(a)$$

Question 11

If $$f(x) = x^2$ e$$^{-3x}$$, why is e$$^{-3x}$$ never zero?

Accepted Answer

The exponential function is never zero for any real $$x$$

Question 12

Which of the following is the correct derivative for $$f(x) = x^2$ \ln x$$?

Accepted Answer

$$f'(x) = 2x \ln x + x$$

Question 13

Suppose you want to estimate $$\sqrt[4]{85}$$ using the linear approximation at $$a=81. $$What is the estimated value?

Accepted Answer

$$L(85) = 3 + \frac{4}{108}$$

Question 14

If $$f(x) = x$$^{3/5}$$(4-x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{3}{5}x$$^{-2/5}$$(4-x) - x$$^{3/5}$$

Linear Approximation, Critical Numbers, and Absolute Extrema in Calculus

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