$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

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

Accepted Answer

$$4$$

Question 2

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

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 3

For the function $$f(x) = |x|$$, which statement is true about its derivative at $$x=0$$?

Accepted Answer

The derivative does not exist at $$x=0$$ because the left and right derivatives are not equal.

Question 4

If $$f(x) = 3x^2$ + 2x + 1$$, what is $$f'(x)$$?

Accepted Answer

$$6x + 2$$

Question 5

Find the equation of the tangent line to $$f(x) = 3x^2$ + 2x + 1 at$$ $$x = -2.$$

Accepted Answer

$$y = -10x + 15$$

Question 6

Which notation does NOT represent the derivative of $$f(x)$$?

Accepted Answer

$$f(x)'$$

Question 7

Suppose $$f(x) is $$continuous at $$x=a$$ and $$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$ exists. What does this imply?

Accepted Answer

The function $$f(x) is $$differentiable at $$x=a.$$

Question 8

A car's position at time $$t is $$given by $$s(t) = t^2$ + 2t. $$What is the instantaneous velocity at $$t = 3$$ seconds?

Accepted Answer

$$10 m/s$$

Question 9

Given $$f(x) = 2x - 1$$, use the definition of the derivative to find $$f'(x).$$

Accepted Answer

$$2$$

Question 10

If $$\lim_{x 	o a^-} f(x) = L_1$ and $$\lim_{x 	o a^+} f(x) = $$L_2$$ with $$L_1$$ 
eq $$L_2$$, what can be said about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

The limit does not exist at $$x=a.$$

Question 11

Which of the following is true about the left and right derivatives at a point $$x=a$$?

Accepted Answer

The derivative exists at $$x=a$$ only if both left and right derivatives exist and are equal.

Question 12

For $$f(x) = |x|$$, what is the value of the left-hand derivative at $$x=0$$?

Accepted Answer

$$-1$$

Question 13

If $$f(x) = x^3$, what is the slope of the tangent line at x = 1$?

Accepted Answer

$$3$$

Question 14

A function $$f(x) is $$said to be differentiable at $$x=a if$$:

Accepted Answer

Both the left and right derivatives exist and are equal at $$x=a.$$

Question 15

Given $$f(x) = 3x^2$ + 2x + 1$$, what is $$f'(-2)$$?

Accepted Answer

$$-10$$

Question 16

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 17

If $$f(x) is $$not continuous at $$x=a$$, which statement is true about $$f'(a)$$?

Accepted Answer

$$f'(a)$$ does not exist.

Question 18

Suppose $$f(x)$$ models the height of a plant in centimeters over time $$x in $$days, and $$f'(10) = 2. $$What does this mean in practical terms?

Accepted Answer

On day 10, the plant is growing at a rate of $$2 cm $$per day.

Question 19

If $$f(x) = x^2$, use the definition of the derivative to compute f$$'(x)$$.

Accepted Answer

$$2x$$

Introduction to Derivatives and Limits in Calculus

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