Question 1

Given the graph in the first image, which shows the function $$f(x) = x^2$ (red curve), a secant line (blue), and a tangent line (green) at x=1$, what is the slope of the tangent line to f(x$$)$$ at x=1$?

Accepted Answer

$$2$$

Question 2

In the context of the second image, which shows the oil level in a tank over time, what does the slope $$-62.5$$ represent?

Accepted Answer

The rate at which oil is being removed from the tank at 7:11 A.M.

Question 3

If the cost function for producing $$x$$ units is $$C(x) = 2x + 1000$$, what is the economic interpretation of the slope and the $$y-$$intercept?

Accepted Answer

Slope is the marginal cost, $$y-$$intercept is the fixed cost.

Question 4

Which of the following is the correct point-slope form for a line passing through $$(2, -1)$$ with slope $$3$$?

Accepted Answer

$$y + 1 = 3(x - 2)$$

Question 5

Find the slope of the line passing through the points $$(6, -2)$$ and $$(9, -4).$$

Accepted Answer

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

Question 6

Given $$f(x) = x^3$, what is f$$'(x)$$ using the power rule?

Accepted Answer

$$3x^2$$

Question 7

Evaluate $$\lim_{x 	o 3} \frac{x^2$ - 9}{x - 3}.$$

Accepted Answer

$$6$$

Question 8

If $$f(x) = \sqrt{x}$$, what is $$f'(x)$$ for $$x \u003e 0$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 9

Which of the following statements about continuity and differentiability is correct?

Accepted Answer

If a function is differentiable at $$x=a$$, it must be continuous at $$x=a.$$

Question 10

Given $$s(t) = -16t^2$ + 128t + 5$$, what is the velocity function $$v(t)$$?

Accepted Answer

$$-32t + 128$$

Question 11

If $$C(x) = 6x^2$ + 2x + 10$$, what is the marginal cost when $$x = 10$$?

Accepted Answer

$$122$$

Question 12

Suppose $$f(x) = x^2$ + 2x + 2. $$Use the difference quotient to find $$f'(x).$$

Accepted Answer

$$2x + 2$$

Question 13

If $$f(x) = x^3$ + \frac{1}{x}$$, what is $$f''(x)$$?

Accepted Answer

$$6x - 2x$$^{-3}$$

Question 14

A manufacturer’s cost function is $$C(x) = 0.005x^3$ - 0.5x^2$ + 28x + 300. $$What is the marginal cost when $$x = 50$$?

Accepted Answer

$$23.75$$

Question 15

Given $$f(x) = x^2$, what is the average rate of change from x=1$ to x=2$?

Accepted Answer

$$3$$

Question 16

If $$f(x) = x^2$, what is the instantaneous rate of change at x=1$?

Accepted Answer

$$2$$

Question 17

Let $$R(x) be $$the revenue in thousands of dollars from producing $$x$$ units, with $$R(4) = 7$$ and $$R'(4) = -0.5. $$Estimate the additional revenue from increasing production from $$x=4 to$$ $$x=5.$$

Accepted Answer

$$-0.5$$ thousand dollars

Question 18

If $$f(x) = x^2$, what is the equation of the tangent line at x=1$?

Accepted Answer

$$y - 1 = 2(x - 1)$$

Question 19

Which of the following is NOT a reason a function might fail to be differentiable at a point?

Accepted Answer

The function is linear at that point.

Question 20

If $$f(x) = x^2$, what is the average rate of change from x=1$ to x=1.1$?

Accepted Answer

$$2.1$$

Business Calculus Study Notes: The Derivative

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