Question 1

Suppose the position function of an object is $$s(t) = 3t^2$ + 5$$, where $$t is in $$seconds and $$s is in $$meters. What is the average velocity of the object over the interval $$[10, 10.1]$$?

Accepted Answer

$$60.6$$

Question 2

Given $$s(t) = 3t^2$ + 5$$, what is the instantaneous velocity at $$t = 10$$?

Accepted Answer

$$60$$

Question 3

If $$y = f(x)$$ and $$\frac{dy}{dx} = 8$$ when $$x = 3$$, estimate the change in $$y as$$ $$x$$ changes from $$3 to$$ $$3.5.$$

Accepted Answer

$$4$$

Question 4

Find the rate of change of $$y = x^4$ with respect to x$ and evaluate it at x = 2$.

Accepted Answer

$$16$$

Question 5

Let $$p = 100 - q^2$ be the demand function for a product. What is the rate of change of price with respect to quantity when q = 5$?

Accepted Answer

$$-10$$

Question 6

If the total cost function is $$c = 0.1q^2$ + 3$$, what is the marginal cost when $$q = 4$$?

Accepted Answer

$$0.8$$

Question 7

Given the average cost per unit $$\bar{c} = 0.0001q^2$ - 0.02q + 5 + \frac{5000}{q}$$, what is the marginal cost function?

Accepted Answer

$$0.0002q - 0.02 - \frac{5000}{q^2$}$$

Question 8

Suppose a manufacturer sells a product at $$2 per unit. If q$ units are sold, what is the marginal revenue?

Accepted Answer

$$2$$

Question 9

A sociologist models income as $$y = 6x$$^{9/4}$$ + 5900$$ for $$4 \leq x \leq 16$$, where $$x is $$years of education. What is the rate of change of income with respect to education when $$x = 16$$?

Accepted Answer

$$54$$

Question 10

For the cost function $$c = 0.4q^2$ + 4q + 5$$, what is the rate of change of $$c$$ with respect to $$q$$ when $$q = 2$$?

Accepted Answer

$$5.6$$

Question 11

If the average cost per unit is $$\bar{c} = 200$$ when $$20$$ bikes are produced per day, and the marginal cost is $$150$$, what is the approximate total cost of producing $$21$$ bikes per day?

Accepted Answer

$$4,350$$

Question 12

If $$c is $$the total cost of producing $$q$$ units, what is the formula for the average cost per unit?

Accepted Answer

$$\bar{c} = \frac{c}{q}$$

Question 13

If $$c = 100$$ when $$q = 20$$, what is the average cost per unit?

Accepted Answer

$$5$$

Question 14

Suppose $$y = f(x)$$ and $$\frac{dy}{dx} = 10 at$$ $$x = 5. $$Estimate the change in $$y if$$ $$x$$ increases from $$5 to$$ $$6.$$

Accepted Answer

$$10$$

Question 15

If the total revenue function is $$r(q) = 5q^2$, what is the marginal revenue when q = 3$?

Accepted Answer

$$30$$

Question 16

Given $$c(q) = 0.2q^2$ + 2q + 10$$, what is the marginal cost function?

Accepted Answer

$$0.4q + 2$$

Question 17

If $$\frac{dy}{dx} = 12x^2$ for y$$ = $$4x^3$$, what is the instantaneous rate of change at $$x = 1$$?

Accepted Answer

$$12$$

Question 18

If $$c = 0.4q^2$ + 4q + 5$$, what is $$\frac{\Delta c}{\Delta q}$$ over the interval $$[2, 3]$$?

Accepted Answer

$$6.8$$

Question 19

If the marginal cost at $$q = 10 is$$ $$5$$ and the cost to produce $$10$$ units is $$100$$, estimate the cost to produce $$11$$ units.

Accepted Answer

$$105$$

Business Calculus Study Guide: Rates of Change, Marginal Cost, and Marginal Revenue

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