Q: Suppose the demand function for a product is $$p(q) = 80 - 2q$$, where $$p is $$the price and $$q is $$the quantity sold. What is the marginal revenue (MR) when $$q = 10$$?

$$MR = 80 - 4q$$,\ MR(10) = 80 - 4 \times 10 = 40$

Q: A monopolist faces the demand function $$q = 120 - 2p. $$What is the price elasticity of demand $$E at$$ $$p = 30$$?

$$E = -\frac{p}{q} \frac{dq}{dp} = -\frac{30}{60} \times (-2) = 1$$

Question 1

Suppose the demand function for a product is $$p(q) = 80 - 2q$$, where $$p is $$the price and $$q is $$the quantity sold. What is the marginal revenue (MR) when $$q = 10$$?

Accepted Answer

$$MR = 80 - 4q$$,\ MR(10) = 80 - 4 	imes 10 = 40$

Question 2

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 5$$, find all critical points and classify them using the second derivative test.

Accepted Answer

Critical points at $$x=1 ($$local maximum) and $$x=3 ($$local minimum)

Question 3

If $$y = (2x^2$ + 3x) e$$^{x^2}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = (4x + 3) e$$^{x^2}$$ + (2x^2$ + 3x) \cdot 2x e$$^{x^2}$$

Question 4

A company’s total cost function is $$TC(q) = 200 + 5q + 0.2q^2$. What is the marginal cost when q = 20$?

Accepted Answer

$$MC = 5 + 0.2 	imes 20 = 9$$

Question 5

Given $$y = \ln(3x^2$ + 2x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{6x + 2}{3x^2$ + 2x}$$

Question 6

If $$f(x) = x^4$ - 8x^2$ + 16$$, what are the $$x-$$coordinates of the local minima?

Accepted Answer

$$x = 2$$

Question 7

Suppose the total revenue function is $$TR(q) = 500q - 5q^2$ and the total cost function is TC(q) = 100 + 20q$. At what output q$ is profit maximized?

Accepted Answer

$$q = 48$$

Question 8

Given the function $$y = (x^2$ + 1)^5$, find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{dy}{dx} = 5(x^2$ + 1)^4$ \cdot 2x$$

Question 9

If $$y = \frac{x^2$ + 1}{x - 1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{(2x)(x-1) - (x^2$ + 1)(1)}{(x-1)^2$}$$

Question 10

A monopolist faces the demand function $$q = 120 - 2p. $$What is the price elasticity of demand $$E at$$ $$p = 30$$?

Accepted Answer

$$E = -\frac{p}{q} \frac{dq}{dp} = -\frac{30}{60} 	imes (-2) = 1$$

Question 11

If $$f(x) = e$$^{3x^2 + x}$$, what is f$$'(x)$$?

Accepted Answer

$$f'(x) = (6x + 1) e$$^{3x^2 + x}$$

Question 12

Given $$y = \sqrt{4x + 1}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{2}{\sqrt{4x + 1}}$$

Question 13

If $$y = x^2$ e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 2x e$$^{2x}$$ + x^2$ \cdot 2 e$$^{2x}$$ = (2x + 2x^2$) e$$^{2x}$$

Question 14

Suppose $$f(x) = x^3$ + 2x^2$ - 5x + 1. $$What is the third derivative $$f'''(x)$$?

Accepted Answer

$$f'''(x) = 6$$

Question 15

Given the implicit equation $$x^2$$ + $$y^2 = 25$, what is $$\frac{dy}{dx}$$ at the point (3$$, 4)$$?

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y} = -\frac{3}{4}$$

Question 16

A firm’s profit function is $$\pi(q) = -2q^2$ + 40q - 100. $$What is the output $$q$$ that maximizes profit?

Accepted Answer

$$q = 10$$

Question 17

If $$y = \ln(x^2$ + 1)$$, what is the value of $$\frac{dy}{dx}$$ when $$x = 0$$?

Accepted Answer

$$\frac{dy}{dx} = 0$$

Question 18

Suppose $$y = (3x^2$ - 2x + 1)^7$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = 7(3x^2$ - 2x + 1)^6$ (6x - 2)$$

Question 19

If $$f(x) = x^2$ + 4x + 7$$, what is the equation of the tangent line at $$x = 1$$?

Accepted Answer

$$y = 6x + 2$$

Differentiation and Applications in Calculus for Economics

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