As $$h \to 0$$, the slope of the secant line approaches the slope of the tangent line at $$x.$$

Question 1

Given the piecewise function $$f(x) = \begin{cases} x^2$ + 1 & 	ext{if } x  2 \end{cases}$$, what is $$\lim_{x 	o 2^-} f(x)$$?

Accepted Answer

$$5$$

Question 2

Given the same function as above, what is $$\lim_{x 	o 2^+} f(x)$$?

Accepted Answer

$$3$$

Question 3

Does $$\lim_{x 	o 2} f(x)$$ exist for the function $$f(x) = \begin{cases} x^2$ + 1 & x  2 \end{cases}$$?

Accepted Answer

No, because the left and right limits are not equal.

Question 4

A function $$f is $$continuous at $$x = c if $$all the following are true: $$1) \lim_{x 	o c} f(x)$$ exists, $$2) f(c)$$ exists, 3) ________.

Accepted Answer

$$\lim_{x 	o c} f(x) = f(c)$$

Question 5

Is $$f(x) = \frac{x+1}{x+2}$$ continuous at $$x = -1$$?

Accepted Answer

Yes, because the denominator is not zero at $$x = -1.$$

Question 6

Is $$f(x) = \frac{x+1}{x+2}$$ continuous at $$x = -2$$?

Accepted Answer

No, because the denominator is zero at $$x = -2.$$

Question 7

Evaluate $$\lim_{x 	o 2} \frac{x-2}{x^2$-4}. Is $$this a $$\frac{0}{0}$$ indeterminate form?

Accepted Answer

Yes, and the limit exists and equals $$\frac{1}{4}.$$

Question 8

Given two points on the graph of $$f(x)$$: $$(x, f(x))$$ and $$(x+h, f(x+h))$$, what does $$\frac{f(a+h) - f(a)}{h}$$ represent?

Accepted Answer

The average rate of change of $$f(x)$$ over $$[a, a+h]$$

Question 9

What is the connection between the secant line through $$(x, f(x))$$ and $$(x+h, f(x+h))$$ and $$\lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

As $$h 	o 0$$, the slope of the secant line approaches the slope of the tangent line at $$x.$$

Question 10

If $$y = 58$$, what is $$y'$$ and $$y'(2)$$?

Accepted Answer

$$0$$

Question 11

If $$y = 6x$$, what is $$\frac{dy}{dx}$$ and $$\frac{dy}{dx}(2)$$?

Accepted Answer

$$6$$

Question 12

Find $$f'(x) if$$ $$f(x) = 3x^4$ + 6x^7$.

Accepted Answer

$$12x^3$$ + $$42x^6$$

Question 13

Find the equation of the tangent line at $$x = -6$$ for $$f(x) = x^3$ + 2. $$Write your answer in the form $$y = mx + b.$$

Accepted Answer

$$y = 108x + 390$$

Question 14

An object moves along the y-axis so that its position at time $$t is$$ $$y(t) = 9t^3$ - 9t^2$ + t + 7. $$What is the velocity at $$t = 3$$ seconds?

Accepted Answer

$$217$$

Question 15

Given the demand equation $$p = 14 - \frac{x}{1000}$$ and the cost equation $$C(x) = 7000 + 4x$$, what is the revenue equation?

Accepted Answer

$$R(x) = x \left(14 - \frac{x}{1000}\right)$$

Question 16

What is the profit equation for the Squirtle shells business?

Accepted Answer

$$P(x) = x \left(14 - \frac{x}{1000}\right) - (7000 + 4x)$$

Question 17

What is the marginal profit equation for the Squirtle shells business?

Accepted Answer

$$P'(x) = 14 - \frac{2x}{1000} - 4$$

Question 18

Find the marginal profit at a production level of $$x = 3000$$ for the Squirtle shells business and interpret the result.

Accepted Answer

The marginal profit is $$4$$, meaning producing one more unit at $$x = 3000$$ increases profit by 4 Pokémon dollars.

Question 19

In the context of differentials, what does $$\Delta y$$ represent in the diagram?

Accepted Answer

The actual change in $$f(x) as$$ $$x$$ increases by $$\Delta x$$

Question 20

In the context of differentials, what does $$dy$$ represent in the diagram?

Accepted Answer

The change in $$y$$ predicted by the tangent line as $$x$$ increases by $$dx$$

Question 21

When would we have $$\Delta y \approx dy$$?

Accepted Answer

When $$\Delta x is $$very small

Question 22

Why is $$\Delta y = dy$$ whenever $$f(x) is a $$linear function?

Accepted Answer

Because the tangent line and the function coincide everywhere for linear functions.

Business Calculus Practice Test Guidance: Limits, Continuity, Derivatives, and Applications

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