Question 1

Given the graph above, which of the following statements is true about the relationship between the red curve and the blue line at the point (1, 4)?

Accepted Answer

The blue line is tangent to the red curve at (1, 4).

Question 2

If the red curve represents $$f(x) = x^2$ + 2$$ and the blue line represents $$y = 3x + 1$$, what is the slope of the tangent to the red curve at $$x = 1$$?

Accepted Answer

$$2$$

Question 3

Which equation best represents the blue line shown in the graph if it is tangent to the red curve $$f(x) = x^2$ + 2 at$$ $$x = 1$$?

Accepted Answer

$$y = 2x + 2$$

Question 4

Suppose the red curve is $$f(x) = x^2$ + 2. $$What is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = 4x - 2$$

Question 5

If the blue line is tangent to the red curve at $$x = a$$, what must be true about their slopes at $$x = a$$?

Accepted Answer

The slope of the blue line equals the derivative of the red curve at $$x = a.$$

Question 6

Given $$f(x) = x^2$ + 2$$, use the limit definition to find $$f'(a).$$

Accepted Answer

$$f'(a) = 2a$$

Question 7

If the blue line is $$y = mx + b$$ and is tangent to $$f(x) = x^2$ + 2 at$$ $$x = 1$$, what is the value of $$m$$?

Accepted Answer

$$2$$

Question 8

Which of the following best describes the geometric meaning of the derivative at a point?

Accepted Answer

It is the slope of the tangent line to the curve at that point.

Question 9

If the red curve is $$f(x) = x^2$ + 2$$, what is the y-coordinate of the point of tangency when $$x = 1$$?

Accepted Answer

$$3$$

Question 10

Suppose the blue line is tangent to the red curve at $$x = 1. $$Which of the following is the correct equation for the tangent line?

Accepted Answer

$$y = 2x + 1$$

Calculus I: Derivatives – Definitions, Rules, and Applications

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