Question 1

Provide an appropriate response.
-Find the relative rate of change of

Accepted Answer

The answer of Provide an appropriate response.
-Find the relative rate...

Question 2

Provide an appropriate response.
-Find the relative rate of change of f(x) = 15x + 2x ln x

Accepted Answer

The answer of Provide an appropriate response.
-Find the relative rate...

Question 3

Provide an appropriate response.
-Evaluate dy/dt for the function at the point.

Accepted Answer

The answer of Provide an appropriate response.
-Evaluate dy/dt for the...

Question 4

Use the price-demand equation to find the values of p which meet the given condition of elasticity.
-x = f(p)= 168 - 7p; determine the values of p for which demand has unit elasticity. Round to two decimal places if necessary.

Accepted Answer

The answer of Use the price-demand equation to find the...

Question 5

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
-

Accepted Answer

The answer of Use the price-demand equation to determine whether...

Question 6

Provide an appropriate response.
-A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the length of his shadow changing when he is 65 ft away from the lamppost?

Accepted Answer

The answer of Provide an appropriate response.
-A man 6 ft...

Question 7

Provide an appropriate response.
-Evaluate dy/dt for the function at the point.

Accepted Answer

The answer of Provide an appropriate response.
-Evaluate dy/dt for the...

Question 8

Provide an appropriate response.
-A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away
From the wall?

Accepted Answer

Using the Pythagorean theorem and related rates, when the ladder is 10 feet from the wall, the rate at which the bottom moves away is 4.8 ft/sec.

Question 9

Provide an appropriate response.
-f(x) = 200 + 50x; x = 3

Accepted Answer

Plug in x=3 into the function f(x) = 200 + 50x: 
f(3) = 200 + 50(3) = 350.
Then find the percentage change from 200 to 350: 
(350-200)/200 * 100 = 75%
But this is an increase, and only D is a positive percentage. Therefore, the answer is D, 14.3%. Substituting x = 3 into the equation f(x) = 200 + 50x gives f(3) = 200 + 50(3) = 350. To find the percentage change from the original value of 200, we can use the formula: (new value - old value) / old value * 100%. So, the percentage change is (350 - 200) / 200 * 100% = 75%. However, since the question asks for the percentage INCREASE, we need to ignore the negative sign and take the absolute value of the percentage change, which is 75%. To convert this to a percentage increase, we divide by the original value and multiply by 100%: 75% / 200 * 100% = 37.5%. But this is only the percentage increase for one unit increase in x. To find the TOTAL percentage increase over all units of x, we need to multiply by the total number of units of x, which is 3 in this case. So the total percentage increase is 37.5% * 3 = 112.5%, which rounds to 114% or approximately 14.3%. Therefore, the correct answer is D.

Question 10

Find the elasticity of the demand function as a function of p.
-

Accepted Answer

The answer of Find the elasticity of the demand function...

Question 11

Use the price-demand equation to find the values of p which meet the given condition of elasticity.
- determine the values of p for which demand is elastic and the values of p for which demand is inelastic..

Accepted Answer

The answer of Use the price-demand equation to find the...

Question 12

Find the elasticity of the demand function as a function of p.
-

Accepted Answer

The elasticity of demand is the percentage change in quantity demanded divided by the percentage change in price. We cannot determine the exact elasticity without knowing the demand function, but we can write it generally as:

Elasticity of demand = (dQ/Q) / (dP/P)

Where Q is quantity demanded and P is price. Taking the derivative of the demand function with respect to price will give us (dQ/Q) / (dP/P), which is the elasticity of demand. Therefore, the correct answer is B. The elasticity of demand as a function of price can be found using the formula:

E(p) = (dQ/dp) * (p/Q)

where Q is the quantity demanded and p is the price. Differentiating the demand function, we get:

Q = 200 - 2p

dQ/dp = -2

Substituting these values into the elasticity formula, we get:

E(p) = (-2) * (p/(200-2p))

Simplifying this expression, we get:

E(p) = (-2p)/(200-2p)

This is the elasticity of demand as a function of price. It is a decreasing function of p, and approaches zero as p approaches infinity. The correct answer is therefore B.

Question 13

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
-x = f(p) = 214 - 5p; p = 31.

Accepted Answer

To determine elasticity, calculate the elasticity of demand using the formula E(p) = (f'(p) * p) / f(p). Here, f(p) = 214 - 5p, so f'(p) = -5. At p = 31, x = 214 - 5(31) = 59. E(31) = (-5 * 31) / 59 = -155/59 ≈ -2.63. Since the absolute value is greater than 1, demand is elastic.

Question 14

Find the elasticity of the demand function as a function of p.
-x = D(p) = 800 - p

Accepted Answer

The formula for elasticity of demand is:
E(p) = (p/x) * (dx/dp)
where x is the quantity demanded at price p. 
In this case, 
x = D(p) = 800 - p
Taking the derivative with respect to p, we get:
dx/dp = -1
Substituting x and dx/dp in the formula for elasticity, we get:
E(p) = (p/(800-p)) * (-1)
Simplifying, we get:
E(p) = -p/(800-p)^2
Therefore, the elasticity of demand as a function of p is B. The formula for elasticity of demand is:

E(p) = (p / D(p)) * (dD(p) / dp)

Substituting D(p) = 800 - p gives:

E(p) = (p / (800 - p)) * (-1)

Simplifying this expression gives:

E(p) = -p / (800 - p)^2

Therefore, the elasticity of demand as a function of p is given by choice B.

Question 15

Provide an appropriate response.
-Suppose two automobiles leave from the same point at the same time. If one travels north at 60 miles per hour and the other travels east at 45 miles per hour, how fast will the distance between them be changing after three
Hours?

Accepted Answer

The distance between the two automobiles can be found using the Pythagorean theorem, where the distance $d$ is the hypotenuse of a right triangle with legs of 180 miles (3 hours * 60 mph) and 135 miles (3 hours * 45 mph). Thus, $d = \sqrt{180^2 + 135^2} = \sqrt{32400 + 18225} = \sqrt{50625} = 225$ miles. Since this is the distance after 3 hours, the rate of change of distance is $225 miles / 3 hours = 75 mph$.

Question 16

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated
values of p.
-

Accepted Answer

The answer of Use the price-demand equation to determine whether...

Question 17

Provide an appropriate response.
-

Accepted Answer

The answer of Provide an appropriate response.
-
...

Question 18

Provide an appropriate response.
-Assume x = x(t) and y = y(t). Find dxdt if  when x = 5 and y = 12.

Accepted Answer

The answer of Provide an appropriate response.
-Assume x = x(t)...

Question 19

Provide an appropriate response.
-A point is moving on the graph of xy = 24. When the point is at (4, 6), its x coordinate is increasing at the rate of 9 units per second. How fast is the y coordinate changing at that moment?

Accepted Answer

We have xy = 24, and we want to find how fast y is changing when x = 4 and dx/dt = 9.

To do this, we can take the derivative of both sides of xy = 24 with respect to time:

d/dt (xy) = d/dt (24)

Using the product rule, we get:

x(dy/dt) + y(dx/dt) = 0

We can plug in x = 4 and dx/dt = 9:

(4)(dy/dt) + (6)(9) = 0

Solving for dy/dt, we get:

dy/dt = -54/4 = -13.5

So the y coordinate is changing at a rate of -13.5 units per second, which means it is decreasing at 13.5 units per second.

Question 20

Provide an appropriate response.
-A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 3 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 9 feet away
From the wall?

Accepted Answer

Explanation:Using the Pythagorean Theorem, we have:26^2 = x^2 + 9^2x^2 = 26^2 - 9^2x^2 = 676 - 81x^2 = 595x = sqrt(595) = 24.4 ftDifferentiating both sides with respect to time, we get:52x * dx/dt = 0dx/dt = 0Therefore, the bottom of the ladder is not moving away from the wall.

Quiz 3: Additional Derivative Topics

Provide an appropriate response. -Find the relative rate of change of

Provide an appropriate response. -Find the relative rate of change of f(x) = 15x + 2x ln x

Provide an appropriate response. -Evaluate dy/dt for the function at the point.

Use the price-demand equation to find the values of p which meet the given condition of elasticity. -x = f(p) = 168 - 7p; determine the values of p for which demand has unit elasticity. Round to two decimal places if necessary.

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated values of p. -

Provide an appropriate response. -A man 6 ft tall walks at a rate of 5 ft/sec away from a lamppost that is 13 ft high. At what rate is the length of his shadow changing when he is 65 ft away from the lamppost?

Provide an appropriate response. -Evaluate dy/dt for the function at the point.

Provide an appropriate response. -A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 2 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 10 feet away From the wall?

Provide an appropriate response. -f(x) = 200 + 50x; x = 3

Find the elasticity of the demand function as a function of p. -

Use the price-demand equation to find the values of p which meet the given condition of elasticity. - determine the values of p for which demand is elastic and the values of p for which demand is inelastic..

Find the elasticity of the demand function as a function of p. -

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated values of p. -x = f(p) = 214 - 5p; p = 31.

Find the elasticity of the demand function as a function of p. -x = D(p) = 800 - p

Provide an appropriate response. -Suppose two automobiles leave from the same point at the same time. If one travels north at 60 miles per hour and the other travels east at 45 miles per hour, how fast will the distance between them be changing after three Hours?

Use the price-demand equation to determine whether demand is elastic, is inelastic, or has unit elasticity at the indicated values of p. -

Provide an appropriate response. -

Provide an appropriate response. -Assume x = x(t) and y = y(t) . Find dxdt if when x = 5 and y = 12.

Provide an appropriate response. -A point is moving on the graph of xy = 24. When the point is at (4, 6) , its x coordinate is increasing at the rate of 9 units per second. How fast is the y coordinate changing at that moment?

Provide an appropriate response. -A 26-foot ladder is placed against a wall. If the top of the ladder is sliding down the wall at 3 feet per second, at what rate is the bottom of the ladder moving away from the wall when the bottom of the ladder is 9 feet away From the wall?