Question 1

In the diagram, what does the vertical distance labeled $$dy$$ represent in terms of the function $$y = f(x)$$?

Accepted Answer

$$dy is $$the change in $$y$$ along the tangent line at $$x$$, approximating the change in $$f(x)$$ for a small $$\Delta x.$$

Question 2

Given the diagram, which equation correctly relates $$dy$$, $$f'(x)$$, and $$dx$$?

Accepted Answer

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

Question 3

What does the segment $$\Delta y in $$the diagram represent?

Accepted Answer

$$\Delta y is $$the difference $$f(x + \Delta x) - f(x)$$, the actual change in $$y$$ along the curve.

Question 4

If $$dx = \Delta x$$, which of the following best describes the relationship between $$dy$$ and $$\Delta y$$ for small $$\Delta x$$?

Accepted Answer

$$dy$$ approximates $$\Delta y$$ when $$\Delta x is $$small.

Question 5

In the context of the diagram, what does the point $$Q$$ represent?

Accepted Answer

The point on the tangent line at $$x$$ corresponding to $$x + \Delta x.$$

Question 6

Which of the following statements is true regarding the tangent line shown in the diagram?

Accepted Answer

The tangent line approximates the curve near the point $$P.$$

Question 7

Suppose $$f(x) is $$differentiable at $$x. $$Which of the following best describes the meaning of $$f'(x) in $$the context of the diagram?

Accepted Answer

$$f'(x) is $$the instantaneous rate of change of $$f(x) at$$ $$x$$, i.e., the slope of the tangent line at $$P.$$

Question 8

If $$f(x) = x^2$, x = 1$, and $$\Delta x = 0.1$$, compute dy$ using the differential formula.

Accepted Answer

$$dy = 2 \cdot 1 \cdot 0.1 = 0.2$$

Question 9

For the function $$y = f(x)$$, which of the following best describes the geometric meaning of the differential $$dy$$?

Accepted Answer

$$dy is $$the vertical distance from $$P to$$ $$Q$$ along the tangent line.

Question 10

If $$f(x) is $$concave up at $$x$$, how does $$dy$$ compare to $$\Delta y$$ for a small positive $$\Delta x$$?

Accepted Answer

$$dy \u003c \Delta y$$

Study Notes: Differentiation and Applications for College Algebra

Study Guide - Practice Questions