Math Core

This test covers the mean value theorem, the extreme value theorem and critical points, increasing and decreasing intervals, the first and second derivative tests, concavity, curve sketching and optimization.

Question 1

Find the value of $c$ guaranteed by the mean value theorem for $f(x) = x^2 + 2x$ on $[0, 4]$ .

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

Question 2

A differentiable function $W$ gives the amount of water, in gallons, in a tank at time $t$ hours. $W(1) = 40$ and $W(6) = 15$ . Which statement is guaranteed by the mean value theorem?

Question 3

Find all critical points of $f(x) = x^3 - 27x + 4$ .

Separate answers with commas, e.g. 2, -5

Question 4

Find the absolute maximum value of $f(x) = x^3 - 3x$ on $[0, 3]$ .

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

Question 5

On what intervals is $f(x) = x^3 - 3x^2 - 24x$ increasing? Use strict inequalities, joining intervals with "or".

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 6

The derivative of a function $h$ is $h'(x) = (x - 1)^2(x + 3)(x - 5)$ . At which value(s) of $x$ does $h$ have a relative maximum?

Question 7

Find the relative maximum value of $f(x) = x^2e^x$ . Give the exact value or a decimal rounded to three places.

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

Question 8

On what interval is $f(x) = x^3 + 3x^2 - 9x$ concave down?

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 9

Find the $x$ -coordinates of all inflection points of $f(x) = x^4 - 2x^3$ .

Separate answers with commas, e.g. 2, -5

Question 10

A twice-differentiable function $g$ has $g'(2) = 0$ and $g''(2) = 5$ . Which statement is true?

The graph of $f'$ , the derivative of a function $f$ , is shown below. Use it for the next two questions.

The graph of f′(x) = 4 − x².Open in grapher →

Question 11

Using the graph of $f'$ , at what value of $x$ does $f$ have a relative maximum?

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

Question 12

Using the graph of $f'$ , on what interval is $f$ concave up?

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5

Question 13

An open-top box with a square base must have a volume of $32$ cubic feet. What is the least possible total area of the base and four sides, in square feet?

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

Question 14

A rectangle is inscribed in a semicircle of radius $5$ , with its base on the diameter. What is the largest possible area of the rectangle?

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