Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is the vertex of the parabola?

Accepted Answer

$$(\frac{3}{4}, \frac{31}{8})$$

Question 2

A company’s cost function is $$C(x) = 500 + 20x$$, and its revenue function is $$R(x) = 50x. $$What is the marginal profit when 100 units are produced?

Accepted Answer

$$30$$

Question 3

Find the equation of the line passing through the points $$(2, 5)$$ and $$(4, 13) in $$slope-intercept form.

Accepted Answer

$$y = 4x - 3$$

Question 4

Evaluate the limit: $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 5

Which of the following statements is TRUE about the function $$f(x) = \frac{1}{x-3}$$?

Accepted Answer

The function is discontinuous at $$x = 3$$ because the denominator is zero.

Question 6

Suppose $$f(x) is $$defined as follows:

$$\begin{cases} x^2$ & x \u003c 1 \ 2x + 1 & x \geq 1 \end{cases}$$

Find $$\lim\limits_{x 	o 1^-} f(x).$$

Accepted Answer

$$1$$

Question 7

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$5e^{0.5}$$

Question 8

Which of the following best describes the derivative of a position function $$s(t)$$ with respect to time $$t$$?

Accepted Answer

It gives the velocity at time $$t.$$

Question 9

If $$f(x) = x^3$ - 2x$$, what is $$f'(2)$$?

Accepted Answer

$$8$$

Question 10

A bank offers continuous compounding at an annual rate of $$6\%. If $$you invest $$\2000$, what will your investment be worth after 5$ years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$2000e^{0.3}$$

Question 11

Which of the following functions has a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{2}{x}$$

Question 12

Given $$f(x) = |x|$$, at which point does the derivative NOT exist?

Accepted Answer

$$x = 0$$

Question 13

If the cost function is $$C(x) = 1000 + 5x + 0.01x^2$, what is the marginal cost when x = 200$?

Accepted Answer

$$9$$

Question 14

Find the $$y-$$intercept of the quadratic function $$f(x) = -3x^2$ + 6x + 9.$$

Accepted Answer

$$9$$

Question 15

Evaluate $$\lim\limits_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 16

A car’s position is given by $$s(t) = 4t^2$ + 2t. $$What is the acceleration at any time $$t$$?

Accepted Answer

$$8$$

Question 17

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 18

Suppose $$f(x) = 2x + 1$$ for $$x \u003c 0$$ and $$f(x) = x^2$ for x$$ \geq 0$$. Does $$\lim\limits_{x 	o 0} f(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal.

Question 19

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 20

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$How many units should be produced to maximize profit?

Accepted Answer

$$10$$

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