Q: A hospital’s cost function for treating patients is $$C(x) = 100x + 500$$, and the revenue function is $$R(x) = 150x. $$What is the profit for treating 20 patients?

$$P = R(x) - C(x) = 150 \times 20 - (100 \times 20 + 500) = 500$$

Question 1

A company’s demand function is given by $$p = 100 - 2q$$, where $$p is $$price and $$q is $$quantity. What is the elasticity of demand when $$q = 20$$?

Accepted Answer

$$E = \frac{dq}{dp} \cdot \frac{p}{q} = -2$$

Question 2

Given the curve $$x^2$$ + $$y^2 = 25$, use implicit differentiation to find the slope of the tangent line at the point (3$$, 4)$$.

Accepted Answer

$$m = \frac{dy}{dx} = -\frac{3}{4}$$

Question 3

A function $$f(x) = x^3$ - 6x^2$ + 9x + 1 is $$defined on $$[0, 5]. $$What is the absolute maximum value of $$f(x) on $$this interval?

Accepted Answer

$$f(5) = 11$$

Question 4

Find the linearization $$L(x) of$$ $$f(x) = \sqrt{x} at$$ $$x = 9$$, and use it to approximate $$\sqrt{9.2}.$$

Accepted Answer

$$L(x) = 3 + \frac{1}{6}(x - 9);\ \sqrt{9.2} \approx 3.033$$

Question 5

Given the revenue function $$R(q) = 50q - 2q^2$, what is the marginal revenue when q = 10$?

Accepted Answer

$$MR = 50 - 4q = 10$$

Question 6

A company’s total cost function is $$C(q) = 20q + 100$$, and its total revenue function is $$R(q) = 40q. $$What is the maximum profit and the number of units sold?

Accepted Answer

Maximum profit is unbounded as $$R(q) \u003e C(q)$$ for all $$q \u003e 5$$

Question 7

Use logarithmic differentiation to find $$\frac{d}{dx} y if$$ $$y = x^x.$$

Accepted Answer

$$\frac{d}{dx} y = x^x (1 + \ln x)$$

Question 8

A company’s demand function is $$p = 60 - 3q. $$What price $$p$$ maximizes revenue?

Accepted Answer

$$p = 30$$

Question 9

Given $$y = x^2$ e$$^{3x}$$, use logarithmic differentiation to find $$\frac{dy}{dx}$$.

Accepted Answer

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

Question 10

A hospital’s cost function for treating patients is $$C(x) = 100x + 500$$, and the revenue function is $$R(x) = 150x. $$What is the profit for treating 20 patients?

Accepted Answer

$$P = R(x) - C(x) = 150 	imes 20 - (100 	imes 20 + 500) = 500$$

Question 11

For the function $$f(x) = x^2$ - 4x + 5$$, what is the minimum value on the interval $$[0, 5]$$?

Accepted Answer

$$f(2) = 1$$

Question 12

A company’s daily revenue is given by $$R(q) = 80q - q^2$. What is the marginal revenue when q = 30$?

Accepted Answer

$$MR = 80 - 2 	imes 30 = 20$$

Question 13

Given the curve $$y^2 = 4x$, use implicit differentiation to find $$\frac{dy}{dx}$$ at the point (1$$, 2)$$.

Accepted Answer

$$\frac{dy}{dx} = 1$$

Question 14

A company’s demand function is $$p = 80 - 4q. $$What is the type of elasticity when $$q = 10$$?

Accepted Answer

Unit elastic ($$|E| = 1)$$

Question 15

Find the equation of the tangent line to the curve $$x^2$$ + $$y^2 = 25$ at the point (3$$, 4)$$.

Accepted Answer

$$y - 4 = -\frac{3}{4}(x - 3)$$

Question 16

A company’s total revenue function is $$R(q) = 100q - 5q^2$ and total cost function is C(q) = 20q + 50$. What is the maximum profit and the number of units sold?

Accepted Answer

Maximum profit at $$q = 8$$, profit $$P = 290$$

Question 17

Given $$f(x) = x^2$ + 2x$$, find the linearization at $$x = 1$$ and use it to approximate $$f(1.1).$$

Accepted Answer

$$L(x) = 3 + 4(x - 1);\ f(1.1) \approx 3.4$$

Question 18

A company’s demand function is $$p = 120 - 6q. $$What is the price for maximum revenue?

Accepted Answer

$$p = 60$$

Business Calculus: Differentiation, Applications, and Optimization

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