Q: Section 5.1 introduces the concept of the definite integral. Which of the following best represents the definition of the definite integral of $$f(x)$$ from $$a to$$ $$b$$?

$$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$$

Question 1

Suppose you are studying for Exam 1, which covers sections 4.5 to 4.7. If section 4.5 is about the Mean Value Theorem, which of the following statements best describes the Mean Value Theorem?

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 2

Section 5.1 introduces the concept of the definite integral. Which of the following best represents the definition of the definite integral of $$f(x)$$ from $$a to$$ $$b$$?

Accepted Answer

$$\int_a^b f(x)\,dx = \lim_{n 	o \infty} \sum_{i=1}^n f(x_i^*) \Delta x$$

Question 3

A pharmaceutical company uses the Fundamental Theorem of Calculus (section 5.3) to calculate the total amount of a drug in the bloodstream over time. If $$C(t) is $$the concentration at time $$t$$, which formula gives the total amount from $$t=a to$$ $$t=b$$?

Accepted Answer

$$\int_a^b C(t)\,dt$$

Question 4

Section 6.1 covers the area between curves. If $$f(x) is $$above $$g(x) on$$ $$[a, b]$$, what is the area between the curves?

Accepted Answer

$$\int_a^b [f(x) - g(x)]\,dx$$

Question 5

A biologist models the population growth of bacteria using a separable differential equation (section 7.1). Which of the following is a separable equation?

Accepted Answer

$$\frac{dy}{dt} = ky$$

Question 6

Section 7.2 discusses exponential growth and decay. If a radioactive substance decays at a rate proportional to its mass, which equation models this process?

Accepted Answer

$$\frac{dm}{dt} = -km$$

Question 7

Section 8.1 covers sequences. Which of the following sequences converges?

Accepted Answer

$$a_n = \frac{1}{n}$$

Question 8

Section 8.2 discusses series. Which of the following series converges?

Accepted Answer

$$\sum_{n=1}^\infty \frac{1}{n^2$}$$

Question 9

A medical researcher uses the integral test (section 8.2) to determine if $$\sum_{n=2}^\infty \frac{1}{n \ln n}$$ converges. What is the corresponding improper integral?

Accepted Answer

$$\$$int_2$$^\infty \frac{1}{x \ln x}\,dx$$

Question 10

Section 5.7 covers the substitution rule. Which substitution would you use to evaluate $$\int 2x \cos(x^2$)\,dx$$?

Accepted Answer

Let $$u = x^2$

Question 11

A physicist calculates the work done by a variable force $$F(x) = 3x^2$ from x=0$ to x=2$ (section 6.2). What is the total work done?

Accepted Answer

$$\$$int_0$$^2 3x^2$\,dx$$

Question 12

Section 7.3 covers logistic growth. Which equation models logistic population growth?

Accepted Answer

$$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$

Question 13

Section 8.3 discusses the ratio test. For which of the following series would the ratio test be most useful?

Accepted Answer

$$\sum_{n=1}^\infty \frac{n!}{2^n}$$

Question 14

A chemist uses the concept of average value (section 6.5) to find the average temperature $$T(t) in a $$reaction vessel from $$t=0 to$$ $$t=10. $$Which formula should be used?

Accepted Answer

$$\frac{1}{10-0} \$$int_0$$^{10} T(t)\,dt$$

Question 15

Section 6.6 covers volumes by slicing. Which integral gives the volume of a solid with cross-sectional area $$A(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\int_a^b A(x)\,dx$$

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