Q: Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

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

Q: A bacteria population grows according to $$P(t) = 500e$$^{0.3t}$$, where t$ is in hours. What is the population’s rate of growth at t = 5$ hours?

$$P'(5) = 0.3 \times 500e$$^{1.5}$$

Question 1

Which of the following best describes the primary focus of a first course in calculus, as outlined in the course description?

Accepted Answer

An introduction to the fundamental concepts of limits, continuity, derivatives, and integrals, with applications to real-world problems.

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

$$v(4) = 6(4) + 2 = 26 m/s$$

Question 3

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 4

A medication is administered to a patient and the 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

$$C'(2) = -5e$$^{-1}$$

Question 5

Which of the following statements about the derivative is TRUE?

Accepted Answer

The derivative of a function at a point gives the slope of the tangent line at that point.

Question 6

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

Accepted Answer

$$x = 3$$

Question 7

A company’s profit $$P (in $$thousands of dollars) is given by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold (in thousands). What is the maximum profit, and for how many units sold does it occur?

Accepted Answer

Maximum profit is $$P(3) = 2 at$$ $$x = 3$$

Question 8

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

Accepted Answer

4

Question 9

Which of the following is the correct definition of the derivative of a function $$f at a $$point $$a$$?

Accepted Answer

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

Question 10

A ball is thrown upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$Its height at time $$t$$ seconds is given by $$h(t) = -4.9t^2$ + 20t + 50. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04 s$$

Question 11

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 12

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

Accepted Answer

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

Question 13

A bacteria population grows according to $$P(t) = 500e$$^{0.3t}$$, where t$ is in hours. What is the population’s rate of growth at t = 5$ hours?

Accepted Answer

$$P'(5) = 0.3 	imes 500e$$^{1.5}$$

Question 14

Which of the following best describes the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$[a, b]$$ such that $$f(c) = N.$$

Question 15

A chemical reaction’s rate is given by $$R(t) = 4t^3$ - 6t^2$ + 2t. At $$what time $$t is $$the rate of reaction at a minimum for $$t \geq 0$$?

Accepted Answer

$$t = 1$$

Question 16

If $$f(x) = \sqrt{x}$$, what is the equation of the tangent line to $$f at$$ $$x = 4$$?

Accepted Answer

$$y = \frac{1}{4}x + 1$$

Question 17

Which of the following is the correct application of the chain rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 18

A tank is being filled with water at a rate given by $$R(t) = 5t^2$ liters per minute. How much water is added to the tank between t = 0$ and t = 3$ minutes?

Accepted Answer

$$\$$int_0$$^3 5t^2$ dt = 45$$ liters

Question 19

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

Accepted Answer

$$f'(a) = 0$$ and $$f''(a) \u003c 0$$

Calculus I: Course Overview and Study Guide

Guida di studio - Domande di pratica

Study guide - Flashcard