Q: Find $$\lim_{x \to 5} \frac{x}{x^2$ - 25} if it $$exists.

$$\lim_{x \to 5} \frac{x}{x^2$ - 25} = \text{DNE}$$

Q: Given $$\lim_{x \to 2} \frac{x^2$ + x + 10}{x^2$ - x - 2}$$, what is the value of the limit?

$$\lim_{x \to 2} \frac{x^2$ + x + 10}{x^2$ - x - 2} = 15$$

Question 1

A company models the price-demand relationship for microchips with the function $$p(x) = 75 - 3x$$, where $$p is $$the price per chip (in dollars) and $$x is $$the number of chips sold (in millions). What is the price per chip if the company sells 7 million chips?

Accepted Answer

$$p(7) = 54$$

Question 2

Given the price-demand function $$p(x) = 75 - 3x$$, what is the revenue function $$R(x) in $$terms of $$x$$?

Accepted Answer

$$R(x) = 75x - 3x^2$

Question 3

Using the revenue function $$R(x) = 75x - 3x^2$, what is the revenue when the demand is 11 million chips?

Accepted Answer

Question 4

Suppose the cost of renting a car for $$x$$ days is given by the piecewise function:

$$C(x) = \begin{cases} 50x & 0  7 \end{cases}$$

What is the cost of renting the car for 5 days?

Accepted Answer

$$C(5) = 230$$

Question 5

Given the supply and demand for barley are $$S(p) = 7,500$$ and $$D(p) = 7,900$$ bushels at a price of $$p = 2.82$$ per bushel, what is the equilibrium price if the price-supply and price-demand equations are linear and intersect at this point?

Accepted Answer

$$p = 2.82$$

Question 6

A company manufacturing surfboards has fixed costs of $$300$$ per day and total costs of $$5,100$$ per day at a daily output of $$200$$ boards. What is the average cost per board at this production rate?

Accepted Answer

$$\frac{5,100}{200} = 25.50$$

Question 7

Suppose $$2,500 is $$invested at $$7\%$$ compounded quarterly. How much money will be in the account after $$15$$ years? Use the formula $$A = P\left(1 + \frac{r}{n}\$$right)^{nt}$$.

Accepted Answer

$$A \approx 7,000$$

Question 8

If $$f(x) = -\frac{3}{x^4$} + 4$$ and $$h(x) = x^2$ - 2x + 8$$, which function has a horizontal asymptote, and what is its equation?

Accepted Answer

$$f(x)$$ has a horizontal asymptote at $$y = 4$$

Question 9

Given the graph of $$g(x)$$ shown, what is $$\lim_{x 	o 2^-} g(x)$$?

Accepted Answer

$$\lim_{x 	o 2^-} g(x) = 2$$

Question 10

Find $$\lim_{x 	o 5} \frac{x}{x^2$ - 25} if it $$exists.

Accepted Answer

$$\lim_{x 	o 5} \frac{x}{x^2$ - 25} = 	ext{DNE}$$

Question 11

A mutual fund earns an average annual return of $$9.58\%$$ compounded annually. How long (to the nearest year) will it take for the money invested in this fund to double? Use the formula $$t = \frac{\ln 2}{\ln(1 + r)}.$$

Accepted Answer

$$t \approx 8$$ years

Question 12

Find $$\lim_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 13

Given $$f(x) = x^2$ - 2x + 8$$, what is the vertex of the parabola?

Accepted Answer

$$(1, 7)$$

Question 14

If $$C(x) = 150 + 40(x-3)$$ for $$3 \u003c x \leq 7$$, what is the cost of renting a car for 7 days?

Accepted Answer

$$C(7) = 310$$

Question 15

Find $$\lim_{x 	o 4} \sqrt{x + 4}.$$

Accepted Answer

$$\sqrt{8}$$

Question 16

A company’s cost function for surfboards is $$C(x) = 300 + 24x. $$What is the marginal cost per board?

Accepted Answer

$$24$$

Question 17

Given $$\lim_{x 	o 2} \frac{x^2$ + x + 10}{x^2$ - x - 2}$$, what is the value of the limit?

Accepted Answer

$$\lim_{x 	o 2} \frac{x^2$ + x + 10}{x^2$ - x - 2} = 15$$

Question 18

If $$C(x) is $$the total cost function and $$C(x) is $$linear, what does the slope of $$C(x)$$ represent in a manufacturing context?

Accepted Answer

The marginal cost per unit

Question 19

Suppose the price-supply equation for barley is $$p = m x + b. If $$the supply increases by 1,000 bushels for every $$0.50$$ increase in price, what is the value of $$m$$?

Accepted Answer

$$m = 0.001$$

Calculus Practice Problems: Functions, Limits, and Applications

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