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

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

Question 1

Evaluate the limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 2

Which of the following statements best describes the meaning of the derivative $$f'(a) at$$ $$x = a$$?

Accepted Answer

It is the slope of the tangent line to $$f(x) at$$ $$x = a.$$

Question 3

Given $$f(x) = x^3$ - 3x^2$ + 2$$, find all critical points.

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 4

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

Accepted Answer

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

Question 5

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees that there exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 6

Find the equation of the tangent line to $$y = x^2$ at x = 1$.

Accepted Answer

$$y = 2x - 1$$

Question 7

If $$f(x) = \sin(x)$$, what is $$f''(x)$$?

Accepted Answer

$$-\sin(x)$$

Question 8

A ball is thrown vertically upward with velocity $$v(t) = 20 - 9.8t. At $$what time does it reach its maximum height?

Accepted Answer

$$t = 2.04 s$$

Question 9

Evaluate $$\$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 10

Which of the following is the best interpretation of the definite integral $$\int_a^b f(x) dx$$?

Accepted Answer

The area under the curve $$f(x)$$ from $$x = a to$$ $$x = b$$

Question 11

If $$f(x) = e$$^{2x}$$, what is $$\frac{d}{dx}f(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 12

A tank is being filled with water at a rate $$R(t) = 5t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 4$$ minutes?

Accepted Answer

$$40$$ liters

Question 13

Which of the following is a necessary condition for a function $$f(x) to $$have a local maximum at $$x = c$$?

Accepted Answer

$$f'(c) = 0$$

Question 14

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 15

Suppose $$f(x) is $$continuous on $$[1, 3]$$ and $$f(1) = -2$$, $$f(3) = 4. $$Which theorem guarantees that $$f(x) = 0$$ for some $$x in$$ $$(1, 3)$$?

Accepted Answer

Intermediate Value Theorem

Question 16

Find $$\int x dx.$$

Accepted Answer

$$\frac{1}{2}x^2$ + C$$

Question 17

A medication is administered at a rate $$r(t) = 10e$$^{-0.5t}$$ mg/hr. How much medication is administered in the first 2 hours?

Accepted Answer

$$20(1 - e$$^{-1}$$) mg$$

Question 18

If $$f(x) = x^4$ - 4x^2$, find the x$-coordinates of all local minima.

Accepted Answer

$$x = \pm \sqrt{2}$$

Question 19

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration.

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