Question 1

Evaluate the following limit: $$\lim_{x 	o -3} \frac{x^2$ - 5x - 24}{x^2$ - 9}$$

Accepted Answer

$$\frac{-8}{-6} = \frac{4}{3}$$

Question 2

Find the limit: $$\lim_{x 	o 2} \frac{3x^2$ + x - 14}{x^2$ - 7x + 10}$$

Accepted Answer

$$\frac{-6}{-3} = 2$$

Question 3

Given $$y = \frac{7}{x} + \frac{30}{x^3$} + \frac{9}{x^6$} - \frac{6}{x^4$}$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$-\frac{7}{x^2$} - \frac{90}{x^4$} - \frac{54}{x^7$} + \frac{24}{x^5$}$$

Question 4

Find the derivative of $$f(x) = \frac{1}{x^4$} + \frac{3}{x^2$}.$$

Accepted Answer

$$-4x^{-5}$$ - $$6x^{-3}$$

Question 5

Find the derivative of $$f(x) = 4x^3$ - 2x^4$ + 6x.$$

Accepted Answer

$$12x^2$$ - $$8x^3 + 6$

Question 6

Find the derivative of $$g(x) = (3x^5$ + 2x^3$) \cdot 4x.$$

Accepted Answer

$$60x^5$$ + $$24x^3$$ + $$12x^4$$ + $$8x^2$$

Question 7

Find the derivative of $$h(x) = 5x^4$ - x^5$ + 1.$$

Accepted Answer

$$20x^3$$ - $$5x^4$$

Question 8

A company models its profit with $$f(x) = x^2$ - 2x$$ where $$x is $$the number of units sold. Find the equation of the tangent line to $$f(x) at $$the point $$(4, 8).$$

Accepted Answer

$$y = 6x - 16$$

Question 9

Consider the piecewise function $$f(x) = \begin{cases} x + 1 & 	ext{if } x  4 \end{cases}. $$What is $$\lim_{x 	o 4^-} f(x)$$?

Accepted Answer

$$5$$

Question 10

For the same piecewise function $$f(x) = \begin{cases} x + 1 & 	ext{if } x  4 \end{cases}$$, what is $$\lim_{x 	o 4^+} f(x)$$?

Accepted Answer

$$9$$

Question 11

For the function $$f(x) = \begin{cases} x + 1 & 	ext{if } x  4 \end{cases}$$, what is $$f(4)$$?

Accepted Answer

undefined

Question 12

Using the limit definition, find the derivative of $$f(x) = x^2$ - 3x + 8.$$

Accepted Answer

$$2x - 3$$

Question 13

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 5t^4$ - t^5$ + 1$$, where $$t is $$time in hours. What is the rate of change of concentration at $$t = 2$$?

Accepted Answer

$$20 \cdot 2^3$ - 5 \cdot 2^4$ = 160 - 80 = 80$$

Question 14

A business models its revenue as $$R(x) = 4x^3$ - 2x^4$ + 6x$$, where $$x is $$the number of products sold. What is the marginal revenue when $$x = 1$$?

Accepted Answer

$$12 \cdot 1^2$ - 8 \cdot 1^3$ + 6 = 12 - 8 + 6 = 10$$

Question 15

A company models its cost function as $$C(x) = \frac{1}{x^4$} + \frac{3}{x^2$}$$, where $$x is $$the number of units produced. What is the marginal cost when $$x = 2$$?

Accepted Answer

$$-4 \cdot 2$$^{-5}$$ - 6 \cdot 2$$^{-3}$$ = -4 \cdot \frac{1}{32} - 6 \cdot \frac{1}{8} = -\frac{1}{8} - \frac{3}{4} = -\frac{7}{8}$$

Question 16

A company models its profit as $$P(x) = 3x^5$ + 2x^3$, and wants to maximize profit by increasing production. If production increases at a rate of 4x$ units per day, what is the rate of change of profit with respect to time?

Accepted Answer

$$(15x^4$$ + $$6x^2$$) \cdot 4x = $$60x^5$$ + $$24x^3$$

Question 17

A company models its revenue as $$R(x) = 7x$$^{-1}$$ + 30x$$^{-3}$$ + 9x$$^{-6}$$ - 6x$$^{-4}$$, where x$ is the number of units sold. What is the marginal revenue when x = 1$?

Accepted Answer

$$-7 - 90 - 54 + 24 = -127$$

Business Calculus Exam 1 Review – Step-by-Step Guidance

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