Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Question 1

Which of the following best describes the difference between the slope of a secant line and the slope of a tangent line to a curve at a point $$x = a$$?

Accepted Answer

The secant line gives the average rate of change over an interval, while the tangent line gives the instantaneous rate of change at a point.

Question 2

Given $$f(x) = x^2$, what is the slope of the tangent line at x = 3$?

Accepted Answer

$$6$$

Question 3

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 4

If $$f(x) = 3x^2$ - 4x + 1$$, what is the equation of the tangent line at $$x = 2$$?

Accepted Answer

$$y = 8x - 9$$

Question 5

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

Accepted Answer

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

Question 6

Given $$f(x) = x^3$ - 2x^2$ + x$$, find $$f'(x).$$

Accepted Answer

$$3x^2 - 2x + 1$

Question 7

Suppose $$f(x) is $$continuous everywhere but has a sharp corner at $$x = 2. $$Which of the following is true?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 2$$

Question 8

If $$f(x) = x^2$ + 2x$$, what is the average rate of change of $$f(x)$$ from $$x = 1 to$$ $$x = 4$$?

Accepted Answer

$$8$$

Question 9

Given the graph of $$f(x)$$, at which points does $$f'(x)$$ cross the $$x-$$axis?

Accepted Answer

Where $$f(x)$$ has a local maximum or minimum

Question 10

A function $$f(x) is $$given by $$f(x) = |x|. $$Which of the following is true about its differentiability at $$x = 0$$?

Accepted Answer

$$f(x) is $$continuous but not differentiable at $$x = 0$$

Question 11

If $$f(x) = x^2$ and g(x) = x + 1$, what is the derivative of h(x) = f(x)g(x$$)$$ using the product rule?

Accepted Answer

$$h'(x) = 2x(x+1) + (x^2$)(1)$$

Question 12

Suppose $$f(x) is $$differentiable everywhere and $$f'(x) \u003e 0$$ for all $$x. $$Which of the following must be true?

Accepted Answer

$$f(x) is $$increasing everywhere

Question 13

Given $$f(x) = x^3$ - 3x^2$ + 2x$$, at what $$x-$$values does the tangent line to $$f(x)$$ become horizontal?

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 14

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, which theorem guarantees that there is some $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 15

A function $$f(x) is $$given by $$f(x) = x^2$ for x$$ \u003c 1$$ and f(x) = 2x - 1$ for x$$ \geq 1$$. Is f(x$$)$$ differentiable at x = 1$?

Accepted Answer

No, because the left and right derivatives at $$x = 1$$ are not equal.

Derivatives: Tangent Lines, Differentiability, and Basic Rules

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