Q: Which of the following statements about the limit definition of the derivative is correct?

$$f'(x) = \displaystyle \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Question 1

For what value of $$a is $$the function 
$$
f(x) = \begin{cases} x^2$ + 8x + 12, & x 
eq -6 \ a, & x = -6 \end{cases}
$$
continuous at $$x = -6$$?

Accepted Answer

Question 2

Which of the following is a necessary hypothesis for applying the Intermediate Value Theorem to a function $$f on$$ $$[a, b]$$?

Accepted Answer

$$f$$ must be continuous on $$[a, b]$$

Question 3

Let $$f(x) = \sqrt{3x^2$ + 7x + 5}. To $$use the Intermediate Value Theorem to show $$\sqrt{3x^2$ + 7x + 5} = 3$$ has a solution in $$(0, 1)$$, which of the following steps is correct?

Accepted Answer

Check that $$f(x) is $$continuous on $$[0, 1]$$ and that $$f(0) \u003c 3 \u003c f(1)$$

Question 4

Using the limit definition of the derivative, what is $$f'(x)$$ for $$f(x) = 3x^2$ + 2x - 1$$?

Accepted Answer

$$6x + 2$$

Question 5

Find the derivative of $$f(x) = (3\sqrt{x})^7$.

Accepted Answer

$$\displaystyle 21 \cdot (3\sqrt{x})^6$ \cdot \frac{1}{2\sqrt{x}}$$

Question 6

Compute the derivative of $$f(x) = 7x$$^{11}$$ - 9x^7$.

Accepted Answer

$$77x^{10}$$ - $$63x^6$$

Question 7

Find the derivative of $$f(x) = \cos x \sec x.$$

Accepted Answer

$$	an x$$

Question 8

Let $$f(x) = 9x^2$ - 7x + 5. $$What is the equation of the tangent line to $$y = f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = 29x - 37$$

Question 9

Which of the following best describes the continuity of $$f(x) = \frac{x^2$ - 4x + 3}{x - 3} at$$ $$x = 3$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 3$$ because $$f(3) is $$undefined, but the limit exists.

Question 10

Suppose $$f(x) = \frac{\sqrt{x}}{4} - 9x^5$ - e$$^{x^2}$$. What is f$$'(x)$$?

Accepted Answer

$$\displaystyle \frac{1}{8\sqrt{x}} - 45x^4$ - 2x e$$^{x^2}$$

Question 11

A chemical reaction's rate is modeled by $$R(t) = 7t^2$ - 4t + 1. $$What is the instantaneous rate of change at $$t = 3$$ seconds?

Accepted Answer

$$34$$

Question 12

If $$f(x) = \cos x (7x^2$ - x$$^{7/2}$$)$$, what is $$f'(x)$$?

Accepted Answer

$$-\sin x (7x^2$ - x$$^{7/2}$$) + \cos x (14x - \frac{7}{2}x$$^{5/2}$$)$$

Question 13

Which of the following statements about the limit definition of the derivative is correct?

Accepted Answer

$$f'(x) = \displaystyle \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 14

A population grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t = 0$?

Accepted Answer

$$5$$

Question 15

Which of the following is NOT a step in the continuity checklist for a function $$f at$$ $$x = a$$?

Accepted Answer

Check that $$f'(a)$$ exists.

Question 16

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = 0.$$

Question 17

If $$f(x) = \frac{9x^2$ - 7x + 5}{x^3$ + 6}$$, what is $$f'(x)$$?

Accepted Answer

$$\displaystyle \frac{(18x - 7)(x^3$ + 6) - (9x^2$ - 7x + 5)(3x^2$)}{(x^3$ + 6)^2$}$$

Question 18

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$-e^{-0.4}$$

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