Question 1

In the diagram, what does the vertical distance labeled $$dy$$ represent?

Accepted Answer

The change in $$y$$ along the tangent line at $$x$$

Question 2

Given the function $$y = f(x)$$, what is the formula for the differential $$dy at$$ $$x$$?

Accepted Answer

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

Question 3

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

Accepted Answer

The actual change in $$y$$ along the curve as $$x$$ increases by $$\Delta x$$

Question 4

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

If $$y = f(x)$$ and $$f'(x) = 3x^2$, what is the differential dy$ when x = 2$ and dx = 0.1$?

Accepted Answer

$$dy = 3 \cdot 2^2$ \cdot 0.1 = 1.2$$

Question 6

Suppose $$f(x) = x^3$. Using the linear approximation at x = 1$, estimate f(1.05$$)$$.

Accepted Answer

$$f(1.05) \approx 1 + 3 \cdot 1^2$ \cdot 0.05 = 1.15$$

Question 7

Which of the following statements is true regarding the tangent line at point $$P in $$the diagram?

Accepted Answer

The tangent line approximates the curve near $$x$$

Question 8

If $$f(x) = \sin x$$, what is the linear approximation of $$f(x) at$$ $$x = 0$$?

Accepted Answer

$$L(x) = x$$

Question 9

In the diagram, what does the horizontal distance $$dx = \Delta x$$ represent?

Accepted Answer

The change in $$x$$ from $$x to$$ $$x + \Delta x$$

Question 10

Why is the differential $$dy$$ used in linear approximation?

Accepted Answer

Because it approximates the change in $$y$$ for small $$\Delta x$$

Study Notes: Differentiation and Applications for College Algebra

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