Question 1

Which of the following best describes a point on a graph where the curve is increasing and curving up?

Accepted Answer

The first derivative is positive and the second derivative is positive at that point.

Question 2

Given a function $$f(x)$$, what does it mean if $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = a$$?

Accepted Answer

The function has a local maximum at $$x = a.$$

Question 3

Suppose you are sketching a qualitative graph using sign data. What is the first step you should take?

Accepted Answer

Mark the vertical lines where the graph changes from increasing to decreasing or vice versa.

Question 4

If a graph is described as 'decreasing and curving down,' which of the following is true about its derivatives?

Accepted Answer

$$f'(x) \u003c 0$$ and $$f''(x) \u003c 0$$

Question 5

Which of the following statements about the slope of a line is correct?

Accepted Answer

If the slope is positive, the line increases from left to right.

Question 6

Given the formula for the slope of a line as $$	ext{slope} = \frac{b}{a}$$, what does $$a$$ represent in the context of the graph?

Accepted Answer

$$a is $$the horizontal change (run) between two points.

Question 7

If the slope graph of a function $$f(x) is $$always positive, what can you conclude about $$f(x)$$?

Accepted Answer

$$f(x) is $$always increasing.

Question 8

A car's velocity is given by a function $$v(t). If$$ $$v(t) is $$negative and decreasing, what does this mean about the car's motion?

Accepted Answer

The car is moving backward and speeding up.

Question 9

Which of the following best describes the area under a curve that lies below the x-axis between $$x = a$$ and $$x = b$$?

Accepted Answer

The area is considered negative.

Question 10

If you are given the sign data for a graph, what can you construct from it?

Accepted Answer

A qualitative sketch of the graph that reflects the sign data.

Question 11

Which of the following is NOT a correct sign convention for measuring lengths in area calculations?

Accepted Answer

Length measured left from the reference point is positive.

Question 12

Suppose a function $$f(x) is $$increasing for $$x  2. $$What can you say about $$x = 2$$?

Accepted Answer

$$x = 2 is a $$local maximum.

Question 13

In the context of calculus, what does the 'slope graph' of a function represent?

Accepted Answer

A graph whose height at each $$x is $$the slope of the original function at $$x.$$

Question 14

If a function $$f(x) is $$flat at $$x = a$$, which of the following is true?

Accepted Answer

$$f'(a) = 0$$

Question 15

A graph is described as 'curving up.' What does this mean in terms of the second derivative?

Accepted Answer

$$f''(x) \u003e 0$$

Question 16

Which of the following real-world scenarios best illustrates the concept of a negative area under a curve?

Accepted Answer

The net change in a bank account balance, where withdrawals are below the x-axis.

Question 17

If a function $$f(x) is $$always positive and its slope is always negative, which of the following is true?

Accepted Answer

$$f(x) is $$decreasing and above the x-axis.

Question 18

Which of the following is a correct interpretation of the area between a curve and the x-axis from $$x = a to$$ $$x = b$$?

Accepted Answer

It represents the net signed area, counting areas below the x-axis as negative.

Question 19

When sketching a graph using 'basic noodles,' what is a 'noodle joint'?

Accepted Answer

A point where one basic noodle ends and another begins, marked with a perpendicular line.

Calculus Study Notes: Graphs, Slope, and Area

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