Q: Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 4$$ seconds?

$$v(4) = 3 \times 2 \times 4 + 2 = 26 m/s$$

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

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

Q: A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. How fast is the population growing at t = 10$ hours?

$$P'(10) = 100 \times 0.05 \times e$$^{0.5}$$ = 5e$$^{0.5}$$

Question 1

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

Accepted Answer

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

Question 2

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$v(4) = 3 	imes 2 	imes 4 + 2 = 26 m/s$$

Question 3

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

Accepted Answer

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

Question 4

A company’s profit $$P(x) in $$thousands of dollars is modeled by $$P(x) = -2x^2$ + 12x - 7$$, where $$x is $$the number of units sold in thousands. At what number of units sold is the profit maximized?

Accepted Answer

$$x = 3$$

Question 5

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

Accepted Answer

$$2e^{2x}$$

Question 6

Which of the following statements about the chain rule is correct?

Accepted Answer

If $$y = f(g(x))$$, then $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x).$$

Question 7

A medication is administered to a patient and the concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$ mg/L, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -1e$$^{-0.6}$$ mg/L per hour

Question 8

Given $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 9

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

If $$h(x) = f(x)g(x)$$, then $$h'(x) = f'(x)g(x) + f(x)g'(x).$$

Question 10

A bacteria population grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours. How fast is the population growing at t = 10$ hours?

Accepted Answer

$$P'(10) = 100 	imes 0.05 	imes e$$^{0.5}$$ = 5e$$^{0.5}$$

Question 11

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at which $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 12

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

Accepted Answer

$$2\cos(2x)$$

Question 13

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

Accepted Answer

$$t = 2.04 s$$

Question 14

If $$y = \frac{x^2$ + 1}{x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 15

Which of the following is the correct statement of the Mean Value Theorem for derivatives?

Accepted Answer

If $$f is $$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 16

A chemical reaction’s rate is given by $$R(t) = 4t^3$ - 3t^2$ + 2t$$, where $$t is in $$minutes. At what time is the rate of reaction increasing most rapidly?

Accepted Answer

$$t = \infty$$

Question 17

If $$f(x) = \sqrt{x^2$ + 4}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{x}{\sqrt{x^2$ + 4}}$$

Question 18

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

Accepted Answer

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

Question 19

A function $$f(x) is $$continuous on $$[1, 5]$$ and differentiable on $$(1, 5). If$$ $$f(1) = 2$$ and $$f(5) = 10$$, what does the Mean Value Theorem guarantee?

Accepted Answer

There exists $$c \in (1, 5)$$ such that $$f'(c) = 4$$

Calculus I Syllabus and Course Structure Overview

Guida di studio - Domande di pratica

Study guide - Flashcard