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A ball's position is $s(t) = 4t^2 + 9t$ meters after $t$ seconds. Find its average velocity on $[0, 6]$ , in m/s.
$s(t) = 5t^2$ . Simplify the average velocity on $[3, 3 + h]$ , $\dfrac{s(3 + h) - s(3)}{h}$ for $h \ne 0$ . Enter an expression in $h$ .
$s(t) = 3t^2 + 7t$ meters. The average velocity on $[6, 6 + h]$ simplifies to $43 + 3h$ m/s. Find the instantaneous velocity at $t = 6$ , in m/s.
$f(x) = x^2 + 4x$ . Use $f'(4) = \displaystyle\lim_{h \to 0} \dfrac{f(4 + h) - f(4)}{h}$ to find $f'(4)$ .
Which expression is $f'(2)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(2 + h) - f(2)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(2 + h) - f(2)\big)$
3. $\dfrac{f(2 + 1) - f(2)}{1}$
4. $f(2)$
Which expression is $f'(9)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(9 + h) - f(9)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(9 + h) - f(9)\big)$
3. $\dfrac{f(9 + 1) - f(9)}{1}$
4. $f(9)$
Which expression is $f'(7)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(7 + h) - f(7)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(7 + h) - f(7)\big)$
3. $\dfrac{f(7 + 1) - f(7)}{1}$
4. $f(7)$
Which expression is $f'(4)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(4 + h) - f(4)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(4 + h) - f(4)\big)$
3. $\dfrac{f(4 + 1) - f(4)}{1}$
4. $f(4)$
Which expression is $f'(5)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(5 + h) - f(5)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(5 + h) - f(5)\big)$
3. $\dfrac{f(5 + 1) - f(5)}{1}$
4. $f(5)$
Which expression is $f'(3)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(3 + h) - f(3)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(3 + h) - f(3)\big)$
3. $\dfrac{f(3 + 1) - f(3)}{1}$
4. $f(3)$
Which expression is $f'(8)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(8 + h) - f(8)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(8 + h) - f(8)\big)$
3. $\dfrac{f(8 + 1) - f(8)}{1}$
4. $f(8)$
Which expression is $f'(6)$ ?
1. $\displaystyle\lim_{h \to 0} \dfrac{f(6 + h) - f(6)}{h}$
2. $\displaystyle\lim_{h \to 0} \big(f(6 + h) - f(6)\big)$
3. $\dfrac{f(6 + 1) - f(6)}{1}$
4. $f(6)$
$f(x) = \dfrac{ 4 }{x}$ . Use the limit definition to find $f'(3)$ .
For $f(x) = x^2$ , $f'(x) = 2x$ . Draw the tangent line to the graph at $x = 1$ : place two points on it.
$f(1) = 2$ and $f'(1) = 3$ . Write the tangent line to the graph of $f$ at $x = 1$ : enter the right-hand side of $y = \ldots$
For $f(x) = x^3$ , the difference quotient at $x = -3$ simplifies to $27 + -9h + h^2$ . What is the slope of the tangent line at $x = -3$ ?
Here is the graph of $y = f(x)$ . Tap the point of the graph where $f'(x) = 0$ .
$f(x) = 3x^2 + 8x$ . Use $f'(x) = \displaystyle\lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to find $f'(x)$ .
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 8$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 8 liters per minute.
2. After 2 minutes, the tank holds 8 liters.
3. In the first 2 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 11$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 11 liters per minute.
2. After 7 minutes, the tank holds 11 liters.
3. In the first 7 minutes, 11 liters flowed in.
4. It takes 11 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 19$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 19 liters per minute.
2. After 3 minutes, the tank holds 19 liters.
3. In the first 3 minutes, 19 liters flowed in.
4. It takes 19 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(6) = 7$ . What does this mean?
1. At $t = 6$ min, the volume is increasing at about 7 liters per minute.
2. After 6 minutes, the tank holds 7 liters.
3. In the first 6 minutes, 7 liters flowed in.
4. It takes 7 minutes to add 6 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(9) = 8$ . What does this mean?
1. At $t = 9$ min, the volume is increasing at about 8 liters per minute.
2. After 9 minutes, the tank holds 8 liters.
3. In the first 9 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 9 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 3$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 3 liters per minute.
2. After 4 minutes, the tank holds 3 liters.
3. In the first 4 minutes, 3 liters flowed in.
4. It takes 3 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(6) = 12$ . What does this mean?
1. At $t = 6$ min, the volume is increasing at about 12 liters per minute.
2. After 6 minutes, the tank holds 12 liters.
3. In the first 6 minutes, 12 liters flowed in.
4. It takes 12 minutes to add 6 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 3$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 3 liters per minute.
2. After 2 minutes, the tank holds 3 liters.
3. In the first 2 minutes, 3 liters flowed in.
4. It takes 3 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 20$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 20 liters per minute.
2. After 3 minutes, the tank holds 20 liters.
3. In the first 3 minutes, 20 liters flowed in.
4. It takes 20 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(5) = 7$ . What does this mean?
1. At $t = 5$ min, the volume is increasing at about 7 liters per minute.
2. After 5 minutes, the tank holds 7 liters.
3. In the first 5 minutes, 7 liters flowed in.
4. It takes 7 minutes to add 5 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 6$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 6 liters per minute.
2. After 4 minutes, the tank holds 6 liters.
3. In the first 4 minutes, 6 liters flowed in.
4. It takes 6 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(5) = 16$ . What does this mean?
1. At $t = 5$ min, the volume is increasing at about 16 liters per minute.
2. After 5 minutes, the tank holds 16 liters.
3. In the first 5 minutes, 16 liters flowed in.
4. It takes 16 minutes to add 5 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(9) = 19$ . What does this mean?
1. At $t = 9$ min, the volume is increasing at about 19 liters per minute.
2. After 9 minutes, the tank holds 19 liters.
3. In the first 9 minutes, 19 liters flowed in.
4. It takes 19 minutes to add 9 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(5) = 12$ . What does this mean?
1. At $t = 5$ min, the volume is increasing at about 12 liters per minute.
2. After 5 minutes, the tank holds 12 liters.
3. In the first 5 minutes, 12 liters flowed in.
4. It takes 12 minutes to add 5 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 7$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 7 liters per minute.
2. After 2 minutes, the tank holds 7 liters.
3. In the first 2 minutes, 7 liters flowed in.
4. It takes 7 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 16$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 16 liters per minute.
2. After 3 minutes, the tank holds 16 liters.
3. In the first 3 minutes, 16 liters flowed in.
4. It takes 16 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 13$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 13 liters per minute.
2. After 4 minutes, the tank holds 13 liters.
3. In the first 4 minutes, 13 liters flowed in.
4. It takes 13 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 4$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 4 liters per minute.
2. After 7 minutes, the tank holds 4 liters.
3. In the first 7 minutes, 4 liters flowed in.
4. It takes 4 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(8) = 13$ . What does this mean?
1. At $t = 8$ min, the volume is increasing at about 13 liters per minute.
2. After 8 minutes, the tank holds 13 liters.
3. In the first 8 minutes, 13 liters flowed in.
4. It takes 13 minutes to add 8 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(8) = 4$ . What does this mean?
1. At $t = 8$ min, the volume is increasing at about 4 liters per minute.
2. After 8 minutes, the tank holds 4 liters.
3. In the first 8 minutes, 4 liters flowed in.
4. It takes 4 minutes to add 8 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 15$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 15 liters per minute.
2. After 3 minutes, the tank holds 15 liters.
3. In the first 3 minutes, 15 liters flowed in.
4. It takes 15 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(6) = 19$ . What does this mean?
1. At $t = 6$ min, the volume is increasing at about 19 liters per minute.
2. After 6 minutes, the tank holds 19 liters.
3. In the first 6 minutes, 19 liters flowed in.
4. It takes 19 minutes to add 6 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 11$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 11 liters per minute.
2. After 3 minutes, the tank holds 11 liters.
3. In the first 3 minutes, 11 liters flowed in.
4. It takes 11 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(9) = 14$ . What does this mean?
1. At $t = 9$ min, the volume is increasing at about 14 liters per minute.
2. After 9 minutes, the tank holds 14 liters.
3. In the first 9 minutes, 14 liters flowed in.
4. It takes 14 minutes to add 9 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 9$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 9 liters per minute.
2. After 7 minutes, the tank holds 9 liters.
3. In the first 7 minutes, 9 liters flowed in.
4. It takes 9 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 20$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 20 liters per minute.
2. After 7 minutes, the tank holds 20 liters.
3. In the first 7 minutes, 20 liters flowed in.
4. It takes 20 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 12$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 12 liters per minute.
2. After 4 minutes, the tank holds 12 liters.
3. In the first 4 minutes, 12 liters flowed in.
4. It takes 12 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(8) = 17$ . What does this mean?
1. At $t = 8$ min, the volume is increasing at about 17 liters per minute.
2. After 8 minutes, the tank holds 17 liters.
3. In the first 8 minutes, 17 liters flowed in.
4. It takes 17 minutes to add 8 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 7$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 7 liters per minute.
2. After 3 minutes, the tank holds 7 liters.
3. In the first 3 minutes, 7 liters flowed in.
4. It takes 7 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(9) = 12$ . What does this mean?
1. At $t = 9$ min, the volume is increasing at about 12 liters per minute.
2. After 9 minutes, the tank holds 12 liters.
3. In the first 9 minutes, 12 liters flowed in.
4. It takes 12 minutes to add 9 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 4$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 4 liters per minute.
2. After 2 minutes, the tank holds 4 liters.
3. In the first 2 minutes, 4 liters flowed in.
4. It takes 4 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 5$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 5 liters per minute.
2. After 4 minutes, the tank holds 5 liters.
3. In the first 4 minutes, 5 liters flowed in.
4. It takes 5 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 17$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 17 liters per minute.
2. After 2 minutes, the tank holds 17 liters.
3. In the first 2 minutes, 17 liters flowed in.
4. It takes 17 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 17$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 17 liters per minute.
2. After 7 minutes, the tank holds 17 liters.
3. In the first 7 minutes, 17 liters flowed in.
4. It takes 17 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(9) = 3$ . What does this mean?
1. At $t = 9$ min, the volume is increasing at about 3 liters per minute.
2. After 9 minutes, the tank holds 3 liters.
3. In the first 9 minutes, 3 liters flowed in.
4. It takes 3 minutes to add 9 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 6$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 6 liters per minute.
2. After 2 minutes, the tank holds 6 liters.
3. In the first 2 minutes, 6 liters flowed in.
4. It takes 6 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 11$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 11 liters per minute.
2. After 2 minutes, the tank holds 11 liters.
3. In the first 2 minutes, 11 liters flowed in.
4. It takes 11 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(8) = 18$ . What does this mean?
1. At $t = 8$ min, the volume is increasing at about 18 liters per minute.
2. After 8 minutes, the tank holds 18 liters.
3. In the first 8 minutes, 18 liters flowed in.
4. It takes 18 minutes to add 8 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 10$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 10 liters per minute.
2. After 7 minutes, the tank holds 10 liters.
3. In the first 7 minutes, 10 liters flowed in.
4. It takes 10 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 8$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 8 liters per minute.
2. After 3 minutes, the tank holds 8 liters.
3. In the first 3 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 17$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 17 liters per minute.
2. After 4 minutes, the tank holds 17 liters.
3. In the first 4 minutes, 17 liters flowed in.
4. It takes 17 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 16$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 16 liters per minute.
2. After 4 minutes, the tank holds 16 liters.
3. In the first 4 minutes, 16 liters flowed in.
4. It takes 16 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 4$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 4 liters per minute.
2. After 4 minutes, the tank holds 4 liters.
3. In the first 4 minutes, 4 liters flowed in.
4. It takes 4 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 8$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 8 liters per minute.
2. After 4 minutes, the tank holds 8 liters.
3. In the first 4 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 4 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 12$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 12 liters per minute.
2. After 7 minutes, the tank holds 12 liters.
3. In the first 7 minutes, 12 liters flowed in.
4. It takes 12 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 3$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 3 liters per minute.
2. After 7 minutes, the tank holds 3 liters.
3. In the first 7 minutes, 3 liters flowed in.
4. It takes 3 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 18$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 18 liters per minute.
2. After 7 minutes, the tank holds 18 liters.
3. In the first 7 minutes, 18 liters flowed in.
4. It takes 18 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(8) = 10$ . What does this mean?
1. At $t = 8$ min, the volume is increasing at about 10 liters per minute.
2. After 8 minutes, the tank holds 10 liters.
3. In the first 8 minutes, 10 liters flowed in.
4. It takes 10 minutes to add 8 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 9$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 9 liters per minute.
2. After 3 minutes, the tank holds 9 liters.
3. In the first 3 minutes, 9 liters flowed in.
4. It takes 9 minutes to add 3 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(4) = 7$ . What does this mean?
1. At $t = 4$ min, the volume is increasing at about 7 liters per minute.
2. After 4 minutes, the tank holds 7 liters.
3. In the first 4 minutes, 7 liters flowed in.
4. It takes 7 minutes to add 4 liters.
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(125)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(829)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(819)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(233)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(708)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(217)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(871)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(338)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(120)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(256)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(231)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(735)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(205)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(749)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(350)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(303)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(257)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(485)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(226)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(507)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(799)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(762)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(605)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(886)$ ?
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(693)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(789)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(780)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(617)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(821)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(452)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(339)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(774)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(858)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(534)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(624)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(139)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(822)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(402)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(672)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(210)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(536)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(412)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(306)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(737)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(480)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(853)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(123)$ ?
1. dollars per bike
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$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(275)$ ?
1. dollars per bike
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3. bikes per dollar
$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(241)$ ?
1. dollars per bike
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3. bikes per dollar
A car's position is $s(t)$ meters after $t$ seconds. $s(9) = 83$ and $s(9.5) = 88$ . Estimate $s'(9)$ , in m/s.
Here is the graph of the derivative $y = f'(x)$ . Tap the point of this graph at the $x$ -value where $f$ has an inflection point.
A particle's velocity is $v(t) = t^2 + 4t$ m/s. Its acceleration is $a(t) = v'(t)$ . Find $a(6)$ , in m/s².
Is $f(x) = |x - -1|$ differentiable at $x = -1$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = -1$ .
3. No, because $f$ is not continuous at $x = -1$ .
Is $f(x) = |x - 1|$ differentiable at $x = 1$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = 1$ .
3. No, because $f$ is not continuous at $x = 1$ .
Is $f(x) = |x - -4|$ differentiable at $x = -4$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = -4$ .
3. No, because $f$ is not continuous at $x = -4$ .
Is $f(x) = |x - 3|$ differentiable at $x = 3$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = 3$ .
3. No, because $f$ is not continuous at $x = 3$ .
Is $f(x) = |x - -2|$ differentiable at $x = -2$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = -2$ .
3. No, because $f$ is not continuous at $x = -2$ .
Is $f(x) = |x - 4|$ differentiable at $x = 4$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = 4$ .
3. No, because $f$ is not continuous at $x = 4$ .
Is $f(x) = |x - 0|$ differentiable at $x = 0$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = 0$ .
3. No, because $f$ is not continuous at $x = 0$ .
Is $f(x) = |x - -3|$ differentiable at $x = -3$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = -3$ .
3. No, because $f$ is not continuous at $x = -3$ .
Is $f(x) = |x - 2|$ differentiable at $x = 2$ ?
1. No: the graph has a corner there (slopes −1 and 1).
2. Yes, because $f$ is continuous at $x = 2$ .
3. No, because $f$ is not continuous at $x = 2$ .
$f(x) = x^2$ for $x \le 4$ and $f(x) = mx - 16$ for $x > 4$ . Which value of $m$ makes $f$ differentiable at $x = 4$ ?
$f(2) = 3$ and $f'(2) = 2$ . Write the local linearization $L(x)$ of $f$ at $x = 2$ .
$f(x) = \sqrt{x}$ has $f'(x) = \dfrac{1}{2\sqrt{x}}$ . Use the local linearization at $x = 4$ to estimate $\sqrt{ 6 }$ . Give an exact fraction.
Near $x = 4$ , the graph of $f$ is concave up. Does the local linearization at $x = 4$ over- or underestimate $f(x)$ for $x$ near $4$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 4$ the linearization is exact.
Near $x = 3$ , the graph of $f$ is concave up. Does the local linearization at $x = 3$ over- or underestimate $f(x)$ for $x$ near $3$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 3$ the linearization is exact.
Near $x = 7$ , the graph of $f$ is concave up. Does the local linearization at $x = 7$ over- or underestimate $f(x)$ for $x$ near $7$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 7$ the linearization is exact.
Near $x = 2$ , the graph of $f$ is concave up. Does the local linearization at $x = 2$ over- or underestimate $f(x)$ for $x$ near $2$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 2$ the linearization is exact.
Near $x = 1$ , the graph of $f$ is concave up. Does the local linearization at $x = 1$ over- or underestimate $f(x)$ for $x$ near $1$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 1$ the linearization is exact.
Near $x = 5$ , the graph of $f$ is concave up. Does the local linearization at $x = 5$ over- or underestimate $f(x)$ for $x$ near $5$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 5$ the linearization is exact.
Near $x = 6$ , the graph of $f$ is concave up. Does the local linearization at $x = 6$ over- or underestimate $f(x)$ for $x$ near $6$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 6$ the linearization is exact.
Near $x = 9$ , the graph of $f$ is concave up. Does the local linearization at $x = 9$ over- or underestimate $f(x)$ for $x$ near $9$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 9$ the linearization is exact.
Near $x = 8$ , the graph of $f$ is concave up. Does the local linearization at $x = 8$ over- or underestimate $f(x)$ for $x$ near $8$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 8$ the linearization is exact.

Answers

$33$
$5 h + 30$
$43$
$12$
$\displaystyle\lim_{h \to 0} \dfrac{f(2 + h) - f(2)}{h}$
$- \frac{4}{9}$
$y = 2 x - 1$
$3 x - 1$
$27$
$(-1, 1)$
$6 x + 8$
At $t = 2$ min, the volume is increasing at about 8 liters per minute.
dollars per bike
$10$
$(-1, 4)$
$16$
No: the graph has a corner there (slopes −1 and 1).
$8$
$2 x - 1$
$\frac{5}{2}$
Underestimate: the tangent line lies below the curve.