Math Core

This test covers Riemann sums, the definite integral, accumulation functions, properties of definite integrals, the Fundamental Theorem of Calculus, antiderivatives and integration by substitution.

Question 1

The rate at which rain collects in a barrel is $R(t)$ liters per hour. Selected values are shown.

$t$ (hours)	0	1	3	4	7
$R(t)$ (L/hr)	5	9	12	10	6

Use a trapezoidal sum with the four subintervals in the table to estimate the total rain collected, in liters, from $t = 0$ to $t = 7$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

A function $f$ is positive, decreasing and concave down on $[1, 5]$ . Which of the following approximations of $\displaystyle\int_1^5 f(x)\,dx$ is guaranteed to be an overestimate?

Question 3

Which of the following is equal to $\displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \ln\!\left(2 + \frac{3k}{n}\right) \frac{3}{n}$ ?

Question 4

Use geometry to evaluate $\displaystyle\int_{-1}^{3} \big(2 - |x|\big)\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

The next two questions use the graph of $f$ below, which consists of two line segments on $[0, 6]$ , and the function $g(x) = \displaystyle\int_0^x f(t)\,dt$ .

The graph of f: a segment from (0, 0) up to (2, 2), then a segment from (2, 2) down to (6, −2). It crosses the x-axis at x = 4.Open in grapher →

Question 5

Find $g(6)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

At what value of $x$ does $g$ have a relative maximum?

Question 7

If $\displaystyle\int_1^6 f(x)\,dx = 11$ and $\displaystyle\int_4^6 f(x)\,dx = 5$ , evaluate $\displaystyle\int_4^1 \big(2f(x) + 3\big)\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

Find $\dfrac{d}{dx} \displaystyle\int_1^{x^3} \frac{1}{1 + t^2}\,dt$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 9

Evaluate $\displaystyle\int_0^2 \big(3x^2 - 4x + 1\big)\,dx$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 10

A tank holds 200 gallons of water at time $t = 0$ . Water drains out at a rate of $r(t) = 10 + 2t$ gallons per minute. How many gallons are in the tank at $t = 5$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Find $f(x)$ if $f'(x) = 4x^3 - 6x^2$ and $f(1) = 3$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Question 12

$\displaystyle\int \left(\frac{6}{x} - \frac{1}{\sqrt{1 - x^2}}\right) dx =$

Question 13

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

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

$\displaystyle\int \frac{1}{x^2 - 6x + 10}\,dx =$