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

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

Q: A medication is administered such that its concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.1t}$$. What is the rate of change of the concentration at t = 5$ hours?

$$C'(5) = -0.1 \times 50e$$^{-0.5}$$

Q: A tank is being filled with water at a rate given by $$R(t) = 5 + 2t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 10$$ minutes?

$$\$$int_0$$^{10} (5 + 2t) dt = 5 \times 10 + 2 \times \frac{10^2$}{2} = 50 + 100 = 150$$ liters

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

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

Accepted Answer

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

Question 3

A ball is thrown vertically upward with a velocity $$v(t) = 20 - 9.8t. $$What is the maximum height reached by the ball?

Accepted Answer

When $$v(t) = 0$$, $$t = \frac{20}{9.8}$$, so the maximum height is $$\$$int_0$$^{20/9.8} (20 - 9.8t) dt$$

Question 4

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

Accepted Answer

$$\frac{dy}{dx} = \cos(x^2$) \cdot 2x$$

Question 5

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

Accepted Answer

$$f''(x) = 4e$$^{2x}$$

Question 6

Which of the following integrals represents the area under the curve $$y = x^2$ from x = 1$ to x = 3$?

Accepted Answer

$$\$$int_1$$^3 x^2$ dx$$

Question 7

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

Accepted Answer

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

Question 8

Which of the following statements about the Mean Value Theorem is correct?

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 9

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x) dx.$$

Accepted Answer

$$2$$

Question 10

Which of the following is the correct derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 11

A medication is administered such that its concentration in the bloodstream is modeled by $$C(t) = 50e$$^{-0.1t}$$. What is the rate of change of the concentration at t = 5$ hours?

Accepted Answer

$$C'(5) = -0.1 	imes 50e$$^{-0.5}$$

Question 12

Which of the following is the correct antiderivative of $$f(x) = 4x^3$?

Accepted Answer

$$x^4 + C$

Question 13

If $$y = 	an(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 14

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

Accepted Answer

It relates differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a)$$

Question 15

A tank is being filled with water at a rate given by $$R(t) = 5 + 2t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 10$$ minutes?

Accepted Answer

$$\$$int_0$$^{10} (5 + 2t) dt = 5 	imes 10 + 2 	imes \frac{10^2$}{2} = 50 + 100 = 150$$ liters

Question 16

Which of the following is the correct derivative of $$y = \cos(3x)$$?

Accepted Answer

$$-3\sin(3x)$$

Question 17

Suppose $$f(x) is $$continuous on $$[0, 2]$$ and $$\$$int_0$$^2 f(x) dx = 6. $$What is the average value of $$f(x) on$$ $$[0, 2]$$?

Accepted Answer

$$\frac{1}{2} \$$int_0$$^2 f(x) dx = 3$$

Question 18

Which of the following is the correct integral for the volume of a solid of revolution formed by rotating $$y = x^2$ about the x$-axis from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 \pi (x^2$)^2 dx$

Question 19

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

Accepted Answer

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

Calculus I: Core Topics and Learning Objectives

Guida di studio - Domande di pratica

Study guide - Flashcard