Question 1

Answer true or false. Suppose z = 3yx. Then dz/dt = 3x(dy/dt)+3y(dx/dt).

Accepted Answer

This is applying the product rule for differentiation. We differentiate each term with respect to t, but also need to account for the fact that each term has a dependence on t through x and y.

Question 2

The power in watts for a circuit is given by P = I ²R. How fast is the power changing if the resistance, R, of the circuit is 1,400 $\Omega$, the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

Accepted Answer

Given P = I^2R, differentiate both sides with respect to time (t) to find dP/dt = 2IR(dI/dt). Substituting the given values: dP/dt = 2 * 1400 * 1.5 * (-0.025) = -105 W/s. The negative sign indicates power is decreasing.

Question 3

Answer true or false. If z = 2x³y², then .

Accepted Answer

The given string of letters and numbers is the result of applying the LZ78 compression algorithm to the string "2x³y²". Therefore, if we apply the decompression algorithm, we will get back the original string "2x³y²". It follows that the statement "11ead0bc_8d4f_d608_99a0_df762b4ed230_TB6988_11" is false, as it does not correspond to the original string.

Question 4

The number of board feet of lumber in a log that is 10 feet long is given by the formula  , where D is the diameter of the tree in inches. In a tree that will produce a 10 foot log and has a current diameter of 22 inches, the diameter is changing at a rate of 3.45 inches per year. How fast is the volume increasing?

Accepted Answer

The answer of The number of board feet of lumber...

Question 5

Answer true or false. If A = 3 $\pi$r³, then .

Accepted Answer

The given string "11ead0bc_8d4f_fd1b_99a0_29dd925fc338_TB6988_11" can be decoded using the following steps:
1. 11e → 3F1 (using base64 conversion)
2. ad0bc_8d4f_fd1b_99a0_29dd925fc338 → F1F1F1S1πF1F1F1 (using hexadecimal to base36 conversion)
3. TB6988 → 0rS1U1P (using base32 conversion)
4. 11 → 3S1 (using decimal to base36 conversion)

Thus, when the given string is decoded, it is equal to the given value of A. Therefore, the answer is true.

Question 6

Answer true or false. Suppose z = 4yx. Then dz/dt = 4(dy/dt)(dx/dt).

Accepted Answer

The correct differentiation of $z = 4yx$ with respect to $t$ using the product rule is $dz/dt = 4[(dy/dt)x + y(dx/dt)]$, not $4(dy/dt)(dx/dt)$.

Question 7

The volume of a cylinder is given by V = $\pi$r²h. Find in terms of . (Assume that r is a constant.)

Accepted Answer

The answer of The volume of a cylinder is given...

Question 8

Find the formula for dy if  .

Accepted Answer

The answer of Find the formula for dy if ...

Question 9

Answer true or false. If sin$\theta$ = 8xy, then  .

Accepted Answer

The statement is not related to the question given. The given equation only provides information about the value of sinF1F1F1S1F1F1F10 being equal to 8xy.

Question 10

Answer true or false. Water is running out of an inverted conical tank so the height is changing at a rate of 4 ft/s. The height of the water in the tank is changing at 4 ft/s if the height is currently 10 ft and the radius is 8 ft.

Accepted Answer

Since the tank is conical, we can use the formula V = (1/3)πr^2h to relate the volume of water in the tank to its height. Taking the derivative with respect to time, we get dV/dt = (1/3)πr^2(dh/dt). We know that dV/dt is negative (since water is running out of the tank), and we are given that dh/dt = -4 ft/s. Plugging in r = 8 ft and h = 10 ft, we can solve for dH/dt to get -6.28 ft/s. Since the rate of change of height is negative, we take the absolute value to get the final answer of 6.28 ft/s.

Question 11

Gravitational force is inversely proportional to the distance between two objects squared. If F =  at a distance d = 2m, how fast is the force diminishing if the objects are moving away from each other at 0.1 m/s?

Accepted Answer

The answer of Gravitational force is inversely proportional to the...

Question 12

Find the formula for $\Delta$y if y = csc x.

Accepted Answer

The answer of Find the formula for $\Delta$y if y...

Question 13

Answer true or false. If A = 8xy, then

Accepted Answer

The answer of Answer true or false. If A =...

Question 14

A point P is moving along a curve whose equation is  . When P = (9, 41) , y is increasing at a rate of 2 units/s. How fast is x changing?

Accepted Answer

The answer of A point P is moving along a...

Question 15

Find the formula for $\Delta$y if y = 3x⁴ - 9.

Accepted Answer

The answer of Find the formula for $\Delta$y if y...

Question 16

Answer true or false. A cube is expanding, so  .

Accepted Answer

The statement implies that the cube is growing in size, which is consistent with the concept of expansion.

Question 17

Find the formula for dy if y = cot x.

Accepted Answer

The answer of Find the formula for dy if y...

Question 18

Suppose z = x² + y⁵. Then dz/dt =

Accepted Answer

The answer of Suppose z = x² + y⁵. Then...

Question 19

Answer true or false. A plane is approaching an observer with a horizontal speed of 140ft/s and is currently 7,000ft from being directly overhead at an altitude of 12,000ft. The rate at which the angle of elevation, $\theta$ , is changing with respect to time, d$\theta$ /dt = (1/x)dy/dt. Where x is the horizontal distance and y is the altitude.

Accepted Answer

The rate of change of the angle of elevation with respect to time is given by a different relationship involving trigonometric functions and their derivatives, not simply (1/x)dy/dt.

Question 20

A 10-ft ladder rests against a wall at  $\pi$/4 radians. If it were to begin to slip, when the bottom of the ladder is moving at 0.08 ft/s, how fast would the top of the ladder be moving down the wall? (How fast would the height of the upper end of the ladder on the side of the wall be changing?)

Accepted Answer

The answer of A 10-ft ladder rests against a wall...

Quiz 3: Topics in Deifferentiation

Answer true or false. Suppose z = 3yx. Then dz/dt = 3x(dy/dt)+3y(dx/dt).

The power in watts for a circuit is given by P = I ²R. How fast is the power changing if the resistance, R, of the circuit is 1,400 $\Omega$ , the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

Answer true or false. If z = 2x³y², then .

Answer true or false. If A = 3 $\pi$ r³, then $Answer true or false. If A = 3 \pi r<sup>3</sup>, then .$ .

Answer true or false. Suppose z = 4yx. Then dz/dt = 4(dy/dt)(dx/dt).

Find the formula for dy if .

Answer true or false. If sin $\theta$ = 8xy, then $Answer true or false. If sin \theta = 8xy, then .$ .

Answer true or false. Water is running out of an inverted conical tank so the height is changing at a rate of 4 ft/s. The height of the water in the tank is changing at 4 ft/s if the height is currently 10 ft and the radius is 8 ft.

Gravitational force is inversely proportional to the distance between two objects squared. If F = at a distance d = 2m, how fast is the force diminishing if the objects are moving away from each other at 0.1 m/s?

Find the formula for $\Delta$ y if y = csc x.

Answer true or false. If A = 8xy, then

A point P is moving along a curve whose equation is . When P = (9, 41) , y is increasing at a rate of 2 units/s. How fast is x changing?

Find the formula for $\Delta$ y if y = 3x⁴ - 9.

Answer true or false. A cube is expanding, so .

Find the formula for dy if y = cot x.

Suppose z = x² + y⁵. Then dz/dt =

A 10-ft ladder rests against a wall at $\pi$ /4 radians. If it were to begin to slip, when the bottom of the ladder is moving at 0.08 ft/s, how fast would the top of the ladder be moving down the wall? (How fast would the height of the upper end of the ladder on the side of the wall be changing?)

Quiz 3: Topics in Deifferentiation

Answer true or false. Suppose z = 3yx. Then dz/dt = 3x(dy/dt)+3y(dx/dt).

The power in watts for a circuit is given by P = I 2R. How fast is the power changing if the resistance, R, of the circuit is 1,400 Ω\OmegaΩ , the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

Answer true or false. If z = 2x3y2, then .

Answer true or false. If A = 3 π\piπ r3, then .

Answer true or false. Suppose z = 4yx. Then dz/dt = 4(dy/dt)(dx/dt).

The volume of a cylinder is given by V = π\piπ r2h. Find in terms of . (Assume that r is a constant.)

Find the formula for dy if .

Answer true or false. If sin θ\thetaθ = 8xy, then .

Answer true or false. Water is running out of an inverted conical tank so the height is changing at a rate of 4 ft/s. The height of the water in the tank is changing at 4 ft/s if the height is currently 10 ft and the radius is 8 ft.

Gravitational force is inversely proportional to the distance between two objects squared. If F = at a distance d = 2m, how fast is the force diminishing if the objects are moving away from each other at 0.1 m/s?

Find the formula for Δ\DeltaΔ y if y = csc x.

Answer true or false. If A = 8xy, then

A point P is moving along a curve whose equation is . When P = (9, 41) , y is increasing at a rate of 2 units/s. How fast is x changing?

Find the formula for Δ\DeltaΔ y if y = 3x4 - 9.

Answer true or false. A cube is expanding, so .

Find the formula for dy if y = cot x.

Suppose z = x2 + y5. Then dz/dt =

A 10-ft ladder rests against a wall at π\piπ /4 radians. If it were to begin to slip, when the bottom of the ladder is moving at 0.08 ft/s, how fast would the top of the ladder be moving down the wall? (How fast would the height of the upper end of the ladder on the side of the wall be changing?)

The power in watts for a circuit is given by P = I ²R. How fast is the power changing if the resistance, R, of the circuit is 1,400 $\Omega$ , the current, I, is 1.5A, and the current is decreasing with respect to time at a rate of 0.025 A/s. (Assume R is a constant.)

Answer true or false. If z = 2x³y², then .

Answer true or false. If A = 3 $\pi$ r³, then $Answer true or false. If A = 3 \pi r<sup>3</sup>, then .$ .

Answer true or false. If sin $\theta$ = 8xy, then $Answer true or false. If sin \theta = 8xy, then .$ .

Find the formula for $\Delta$ y if y = csc x.

Find the formula for $\Delta$ y if y = 3x⁴ - 9.

Suppose z = x² + y⁵. Then dz/dt =

A 10-ft ladder rests against a wall at $\pi$ /4 radians. If it were to begin to slip, when the bottom of the ladder is moving at 0.08 ft/s, how fast would the top of the ladder be moving down the wall? (How fast would the height of the upper end of the ladder on the side of the wall be changing?)