Question 1

Given the function $$f(x) = x^3$, what is the value of $$\Delta y$$ when x$ changes from 2 to 2.1?

Accepted Answer

$$\Delta y = 1.261$$

Question 2

Which of the following best describes the difference between $$\Delta y$$ and $$dy$$ for a differentiable function $$y = f(x)$$?

Accepted Answer

$$\Delta y is $$the actual change in $$y$$, while $$dy is $$the approximate change in $$y$$ estimated using the derivative.

Question 3

For the function $$f(x) = x^2$ + 3x$$, what is $$dy$$ when $$x = 2$$ and $$dx = 0.1$$?

Accepted Answer

$$dy = 0.7$$

Question 4

If $$f(x) = x^2$ - 6x$$, what is the value of $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x - 6$$

Question 5

A company’s weekly profit is given by $$P(x) = 8x - 5,000 - x^2$. Use differentials to estimate the change in profit if production increases from 2,000 to 2,010 units per week.

Accepted Answer

$$dP = 40$$

Question 6

Given $$f(x) = x^2$, what is the value of the difference quotient $$\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ when x = 1$ and $$\Delta x = 1$$?

Accepted Answer

$$3$$

Question 7

Which of the following statements is true regarding the secant and tangent lines for a function $$y = f(x)$$?

Accepted Answer

The secant line connects two points on the curve, while the tangent line touches the curve at one point and has slope $$f'(x).$$

Question 8

For $$f(x) = x^2$ - 6x$$, if $$x = 2$$ and $$dx = 0.1$$, what is the value of $$dy$$?

Accepted Answer

$$dy = -1.2$$

Question 9

If $$f(x) = x^2$, what is the approximate change in y$ (dy$) when x$ increases from 1 to 1.2?

Accepted Answer

$$dy = 0.4$$

Question 10

A company’s revenue is given by $$R(x) = 10x - x^2$/500. $$Use differentials to estimate the change in revenue if production increases from 2,000 to 2,010 units per week.

Accepted Answer

$$dR = 60$$

Question 11

Which of the following best describes the use of differentials in business applications?

Accepted Answer

Differentials are used to estimate small changes in cost, revenue, or profit when production levels change slightly.

Question 12

Given $$f(x) = x^2$ - 6x$$, what is $$\Delta y$$ when $$x$$ increases from 2 to 2.1?

Accepted Answer

$$\Delta y = -1.19$$

Question 13

If $$f(x) = x^2$, and $$\Delta x = 0.2$$, what is the actual change in y$ ($$\Delta y$$) when x$ increases from 1 to 1.2?

Accepted Answer

$$\Delta y = 0.36$$

Question 14

Which formula correctly expresses the differential $$dy$$ for a differentiable function $$y = f(x)$$?

Accepted Answer

$$dy = f'(x) dx$$

Question 15

For $$f(x) = x^2$, what is the value of dy$ when x = 1$ and dx = 0.3$?

Accepted Answer

$$dy = 0.6$$

Differentials and Increments in Business Calculus

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