Functions, Derivatives, and Antiderivatives

Reading a function against its own derivative: draining-tank rates, always/sometimes/never statements about slope and concavity, and settling two students' competing claims.

Calculus Lab Manual, Functions, Derivatives, and Antiderivatives

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9 exercises

Exercise 16

As water drains from a tank the drainage rate continually decreases over time. Suppose that a tank that initially holds 360 gal of water drains in exactly $6$ minutes. Answer each of the following questions about this problem situation.

Suppose that $\fe{V}{t}$ is the amount of water (gal)left in the tank where $t$ is the amount of time that has elapsed (s) since the drainage began. Which of the following is the…

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Exercise 17

For each statement in Exercises–this exercise, decide which of the following is true.

The statement is true regardless of the specific function $f$ .
The statement is true for some functions and false for other functions.
The statement is false regardless of the specific function $f$ .

If a function $f$ is increasing over the entire interval $\ointerval{-3}{7}$ and $\fe{\fd{f}}{0}$ exists, then $\fe{\fd{f}}{0}\gt0$ .

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Exercise 18

For each statement in Exercises–this exercise, decide which of the following is true.

The statement is true regardless of the specific function $f$ .
The statement is true for some functions and false for other functions.
The statement is false regardless of the specific function $f$ .

If a function $f$ is increasing over the entire interval $\ointerval{-3}{7}$ and $\fe{\fd{f}}{4}$ exists, then $\fe{\fd{f}}{4}=0$ .

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Exercise 19

For each statement in Exercises–this exercise, decide which of the following is true.

The statement is true regardless of the specific function $f$ .
The statement is true for some functions and false for other functions.
The statement is false regardless of the specific function $f$ .

If a function $f$ is concave down over the entire interval $\ointerval{-3}{7}$ and $\fe{\fd{f}}{-2}$ exists, then $\fe{\fd{f}}{-2}\lt0$ .

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Exercise 20

For each statement in Exercises–this exercise, decide which of the following is true.

The statement is true regardless of the specific function $f$ .
The statement is true for some functions and false for other functions.
The statement is false regardless of the specific function $f$ .

If the slope of $f$ is increasing over the entire interval $\ointerval{-3}{7}$ and $\fe{\fd{f}}{0}$ exists, then $\fe{\fd{f}}{0}\gt0$ .

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Exercise 21

For each statement in Exercises–this exercise, decide which of the following is true.

The statement is true regardless of the specific function $f$ .
The statement is true for some functions and false for other functions.
The statement is false regardless of the specific function $f$ .

If a function $f$ has a local maximum at $3$ , then $\fe{\fd{f}}{3}=0$ .

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Exercise 22

For a certain function $g$ , $\fe{\fd{g}}{t}=-8$ at all values of $t$ on $\ointerval{-\infty}{-3}$ and $\fe{\fd{g}}{t}=2$ at all values of $t$ on $\ointerval{-3}{\infty}$ .

Janice thinks that $g$ must be discontinuous at $-3$ but Lisa disagrees. Who's right?

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Exercise 23

For a certain function $g$ , $\fe{\fd{g}}{t}=-8$ at all values of $t$ on $\ointerval{-\infty}{-3}$ and $\fe{\fd{g}}{t}=2$ at all values of $t$ on $\ointerval{-3}{\infty}$ .

Lisa thinks that the graph of $\sd{g}$ is the line $y=0$ but Janice disagrees. Who's right?

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Exercise 36

Complete as many cells in this table as possible so that when read left to right the relationships between the derivatives, function, and antiderivative are always true. Please note that several cells will remain blank.

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