Q: If $y = x^2$ and $x$ changes from $2$ to $2.01$, use differentials to estimate the change in $y$.

$dy \approx 4 \times 0.01 = 0.04$

Question 1

A 10 ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of $\frac{4}{5}$ ft/s, how fast is the top of the ladder sliding down the wall when the bottom is 6 ft from the wall?

Accepted Answer

$-3$ ft/s

Question 2

The radius of a sphere is increasing at a rate of $2$ cm/min. At what rate is the surface area of the sphere increasing when the radius is $8$ cm?

Accepted Answer

$128\pi$ cm$^2$/min

Question 3

The altitude of a triangle is increasing at a rate of $1$ cm/min while the area is increasing at a rate of $2$ cm$^2$/min. At what rate is the base of the triangle changing when the altitude is $10$ cm and the area is $100$ cm$^2$?

Accepted Answer

$-1.6$ cm/min

Question 4

A cylindrical tank has a constant radius $r$ and height $h(t)$. If the volume of water inside the tank is $V = \pi r^2 h$, what is the rate of change of volume with respect to time?

Accepted Answer

$\frac{dV}{dt} = \pi r^2 \frac{dh}{dt}$

Question 5

For a cone with radius $r$ and height $h$, the volume is $V = \frac{1}{3}\pi r^2 h$. If both $r$ and $h$ are functions of time, what is the correct expression for $\frac{dV}{dt}$?

Accepted Answer

$\frac{dV}{dt} = \frac{1}{3}\pi (2r h \frac{dr}{dt} + r^2 \frac{dh}{dt})$

Question 6

Suppose $y = f(x)$ and $x$ changes from $a$ to $a + dx$. Which of the following best approximates the change in $y$?

Accepted Answer

$dy \approx f'(a) dx$

Question 7

A cube has a side length measured to be $6$ cm with an error of $0.2$ cm. Using differentials, estimate the error in the computed volume of the cube.

Accepted Answer

$21.6$ cm$^3$

Question 8

Which of the following is the correct formula for the differential $dy$ of $y = f(x)$?

Accepted Answer

$dy = f'(x) dx$

Question 9

If the measured value of a quantity is $a + dx$, and the true value is $a$, what does $dx$ represent in the context of error propagation?

Accepted Answer

The measurement error

Question 10

Suppose the radius of a cylinder is constant and the height is increasing at $3$ cm/min. What is the rate of change of the volume if $r = 5$ cm?

Accepted Answer

$75\pi$ cm$^3$/min

Question 11

A function $f(x)$ is approximated near $x = a$ by its tangent line. Which expression gives the linear approximation for $f(a + dx)$?

Accepted Answer

$f(a + dx) \approx f(a) + f'(a) dx$

Question 12

If $y = x^2$ and $x$ changes from $2$ to $2.01$, use differentials to estimate the change in $y$.

Accepted Answer

$dy \approx 4 	imes 0.01 = 0.04$

Question 13

Which step in the problem solving strategy involves expressing the given information and the required rate in terms of derivatives?

Accepted Answer

Step 4

Question 14

In related rates problems, why is it important to assign symbols to all quantities that are functions of time?

Accepted Answer

To correctly apply differentiation with respect to time

Question 15

A tank in the shape of a cone has a radius $r$ and height $h$. If both $r$ and $h$ are increasing, which formula correctly gives the rate of change of the volume?

Accepted Answer

$\frac{dV}{dt} = \frac{1}{3}\pi (2r h \frac{dr}{dt} + r^2 \frac{dh}{dt})$

Question 16

If the side length of a cube is measured as $a$ with an error $dx$, what is the differential $dV$ for the volume $V = a^3$?

Accepted Answer

$dV = 3a^2 dx$

Question 17

Which of the following best describes the use of differentials in error propagation?

Accepted Answer

Differentials estimate how measurement errors affect computed quantities

Question 18

Suppose the surface area $S$ of a sphere is $S = 4\pi r^2$. If $r$ increases by a small amount $dr$, what is the approximate change in $S$?

Accepted Answer

$dS = 8\pi r dr$

Related Rates, Differentials, and Error Propagation in Calculus

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