Q: The percentage error in estimating a value is given by:

$$\text{percentage error} = \text{relative error} \times 100$$

Question 1

A car travels along a straight road and its position at time $$t is $$given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous velocity of the car at $$t = 3$$ seconds?

Accepted Answer

$$v(3) = 26$$

Question 2

The cost $$C(x) to $$produce $$x$$ units of a product is given by $$C(x) = 100 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$MC = 9$$

Question 3

A manufacturer’s average cost function is $$C(x) = \frac{100 + 5x + 0.1x^2$}{x}. $$What is the average cost when 10 units are produced?

Accepted Answer

$$C(10) = 20$$

Question 4

If the position of a particle is given by $$s(t) = t^3$ - 6t^2$ + 9t$$, what is the acceleration of the particle at $$t = 2$$?

Accepted Answer

$$a(2) = 0$$

Question 5

A ball is thrown vertically upward with an initial velocity of $$20 m/s. $$Its height after $$t$$ seconds is $$h(t) = 20t - 4.9t^2$. What is the maximum height the ball will attain?

Accepted Answer

$$h_{max} = 20.41 m$$

Question 6

The total revenue function for selling $$x$$ units of a commodity is $$R(x) = 50x. $$What is the marginal revenue?

Accepted Answer

$$MR = 50$$

Question 7

A factory’s profit function is $$P(x) = -0.01x^2$ + 12x - 100. $$What is the marginal profit when $$x = 50$$?

Accepted Answer

$$MP = 11$$

Question 8

Sand is being dropped onto a conical pile in such a way that the height of the pile is always twice the base radius. If the rate of change of volume of the pile with respect to the radius when the radius is 4 meters is:

Accepted Answer

$$dV/dr = 64\pi$$

Question 9

If $$y = f(x) is a $$differentiable function, the differential $$dy is $$given by:

Accepted Answer

$$dy = f'(x)dx$$

Question 10

A manufacturer’s cost function is $$C(x) = 100 + 5x + 0.1x^2$. What is the actual cost of producing the 21st item?

Accepted Answer

$$C(21) - C(20) = 9.1$$

Question 11

A particle is moving along a horizontal line according to $$s(t) = 4t^2$ - 8t + 2. $$What is the instantaneous velocity at $$t = 2$$?

Accepted Answer

$$v(2) = -2$$

Question 12

The supply equation for a commodity is $$q = 1000 + 2p^2$, where q$ is the units supplied and p$ is the price per unit. If the price is increasing at the rate of 5$ per week when p = 20$, what is the rate of change of supply?

Accepted Answer

$$dq/dt = 400$$

Question 13

If the total cost function is $$C(x) = 100 + 5x + 0.1x^2$, what is the average cost function?

Accepted Answer

$$AC(x) = \frac{100 + 5x + 0.1x^2$}{x}$$

Question 14

A property’s value $$P(t)$$ increases at a rate proportional to its current value. If $$P(0) = 100,000$$ and the rate of increase is $$5\%$$ per year, what is the value after 3 years?

Accepted Answer

$$P(3) = 115,762$$

Question 15

The instantaneous acceleration of a particle at time $$t is $$given by $$a(t) = \frac{dv}{dt}. If$$ $$v(t) = 3t^2$ - 2t + 1$$, what is $$a(2)$$?

Accepted Answer

$$a(2) = 10$$

Question 16

A tank is being filled with water at a rate of $$8 m$^3/$$min. If the volume $$V$$ and height $$h of $$the water are related by $$V = \pi h^2$, what is the rate of change of height when h = 2$ m?

Accepted Answer

$$dh/dt = 2/(4\pi)$$

Question 17

The percentage error in estimating a value is given by:

Accepted Answer

$$	ext{percentage error} = 	ext{relative error} 	imes 100$$

Question 18

A manufacturer’s total profit function is $$P(x) = R(x) - C(x). If$$ $$R(x) = 50x$$ and $$C(x) = 100 + 5x + 0.1x^2$, what is the marginal profit function?

Accepted Answer

$$MP(x) = 50 - 5 - 0.2x$$

Question 19

If $$f(x) is a $$differentiable function, which of the following best describes the geometric meaning of $$f'(x)$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x$$

Rates of Change, Related Rates, and Differentials in Applied Calculus

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