Q: Which of the following limits defines the derivative of $$f(x) at$$ $$x = a$$?

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

Question 1

Which of the following best describes the concept of the derivative at a point $$x = a$$ for a function $$f(x)$$?

Accepted Answer

The slope of the tangent line to the curve $$y = f(x) at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$f'(x) = 3x^2$ - 4x + 1$$

Question 3

Which of the following limits defines the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

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

Question 4

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$ using the chain rule?

Accepted Answer

$$2x \cos(x^2$)$$

Question 5

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

Accepted Answer

$$t = 2.04$$ seconds

Question 6

Which of the following is the correct application of the product rule for differentiation for $$f(x) = u(x)v(x)$$?

Accepted Answer

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

Question 7

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

Accepted Answer

$$y = 4x - 1$$

Question 8

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

Accepted Answer

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

Question 9

Which of the following statements about continuity is true?

Accepted Answer

A function is continuous at $$x = a if$$ $$\displaystyle \lim_{x 	o a} f(x) = f(a).$$

Question 10

Suppose $$f(x)$$ has a local maximum at $$x = c. $$Which of the following must be true?

Accepted Answer

$$f'(c) = 0$$

Question 11

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

Accepted Answer

$$4e^{2x}$$

Question 12

Which of the following is the correct derivative of $$y = \frac{x^2$ + 1}{x}$$ using the quotient rule?

Accepted Answer

$$\frac{x^2$ - 1}{x^2$}$$

Question 13

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = -2x^2$ + 12x - 9$$, where $$x is $$the number of units sold in hundreds. What is the maximum profit?

Accepted Answer

$$15$$ thousand dollars

Question 14

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct application of the chain rule for $$y = \cos(3x)$$?

Accepted Answer

$$y' = -3\sin(3x)$$

Question 16

If $$f(x) = x^4$ - 4x^3$ + 6x^2$, find all critical points.

Accepted Answer

$$x = 0, 1, 2$$

Question 17

Which of the following best describes the Mean Value Theorem for derivatives?

Accepted Answer

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

Question 18

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-3.68$$ units/hour

Question 19

Which of the following is the correct derivative of $$y = 	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 20

If $$f(x) = x^2$ and g(x$$) = \sqrt{x}$$, what is $$\frac{d}{dx}[f(g(x))]$$?

Accepted Answer

$$x^{1/2}$$

MAT 112 Calculus I: Course Outline and Key Topics

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