Q: A function $$f(x) is $$continuous at $$x = a if$$:

$$\lim_{x \to a} f(x)$$ exists, $$f(a)$$ exists, \text{ and } \lim_{x \to a} f(x) = f(a)$

Question 1

A company models its profit with the function $$P(x) = -2x^2$ + 60x + 100$$, where $$x is $$the number of units sold. What is the number of units that should be produced and sold to maximize profit?

Accepted Answer

$$x = 30$$

Question 2

Evaluate the definite integral $$\int_{2}^{5} (3x^2$ - 7x) dx.$$

Accepted Answer

$$71$$

Question 3

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

Accepted Answer

$$12x^3$$ + $$6x^2 + 1$

Question 4

A bakery models its revenue with $$R(x) = 4x$$ and its cost with $$C(x) = 2x + 140. $$What is the break-even point?

Accepted Answer

$$x = 35$$

Question 5

Find the equation of the tangent line to $$y = x^2$ - 3x + 2 at$$ $$x = 1.$$

Accepted Answer

$$y = -1x + 2$$

Question 6

Differentiate $$y = (3x^2$ + 2x$$)^{5}$$ using the Chain Rule.

Accepted Answer

$$y' = 5(3x^2$ + 2x)^4$ \cdot (6x + 2)$$

Question 7

Find the area under the curve $$f(x) = x^2$ from x = 2$ to x = 4$.

Accepted Answer

$$\frac{56}{3}$$

Question 8

Suppose the daily cost, in dollars, of producing $$x$$ violins is $$C(x) = 0.02x^2$ + 2x + 76. $$Use calculus to estimate the daily cost of increasing production from 50 to 52 violins.

Accepted Answer

$$C'(50) 	imes 2$$

Question 9

Find the derivative of $$y = \frac{2x^2$ + 3x}{x}.$$

Accepted Answer

$$y' = \frac{2x + 3}{x} - \frac{2x^2$ + 3x}{x^2$}$$

Question 10

Evaluate $$\int 7x^2$ dx.$$

Accepted Answer

$$\frac{7}{3}x^3$ + c$$

Question 11

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists, $$f(a)$$ exists, 	ext{ and } \lim_{x 	o a} f(x) = f(a)$

Question 12

Find the indicated derivative: $$\frac{d}{dx} (x^3$ - 3x^2$ + 6x - 7).$$

Accepted Answer

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

Question 13

Find the absolute maximum and absolute minimum values of $$f(x) = x^2$ - 4x + 8 on $$the interval $$[2, 6].$$

Accepted Answer

Absolute maximum: $$20$$, absolute minimum: $$4$$

Question 14

Find the antiderivative $$F(x) of$$ $$f(x) = 3x^2$ - 5x$$ given $$F(0) = 8.$$

Accepted Answer

$$F(x) = x^3$ - \frac{5}{2}x^2$ + 8$$

Question 15

Find a simplified difference quotient for $$f(x) = x^2$ + 10.$$

Accepted Answer

$$2x + h$$

Question 16

For the function $$f(x) = x^3$ - 6x^2$ + 9x + 72$$, at which point does the tangent line have slope $$0$$?

Accepted Answer

$$(2, 64)$$

Question 17

Differentiate $$y = \frac{3x^2$ - 5x}{x}.$$

Accepted Answer

$$y' = \frac{3x - 5}{x}$$

Question 18

Find the derivative of $$y = \frac{2x^2$ + 3x}{x}$$ using the quotient rule.

Accepted Answer

$$y' = \frac{2x + 3}{x} - \frac{2x^2$ + 3x}{x^2$}$$

Business Calculus Study Guidance: Limits, Derivatives, Tangents, Optimization, and Integrals

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