Question

1. the graph of a function y = g(x) is shown.

a. is g increasing or decreasing at x = 4?
b. is the rate of change of g increasing or decreasing at x = 4?
c. identify the intervals on which g(x) is increasing.
d. identify the intervals on which g(x) is decreasing.

1. the value of a car with respect to its age can be modeled by the function v.

t (years)	0	1	2	3	4
v(t) ($)	24,000	21,360	19,010	16,919	15,058

a. is the value of the car increasing or decreasing? is the graph of v concave up or concave down? how do you know?

1. at each of the given points of y = g(x), determine if g is increasing or decreasing, and whether the rate of change is increasing or decreasing.

point	g(x)	rate of change
a
b
c
d

1. information of the function l and its rate of change is given in the table below.

x	-∞ < x ≤ - 8	x = - 8	- 8 < x ≤ 2	x = 2	2 ≤ x ≤ 5	x = 5	5 ≤ x < ∞
l	decreasing	5	increasing	10	increasing	21	decreasing
r.o.c.	increasing	0	increasing	12	decreasing	0	decreasing

a. on which interval(s) is the slope of the graph of l positive?
b. on which interval(s) is the graph of l concave down?
c. on which interval(s) are the slopes of l negative and getting flatter?

Explanation:

Step 1: Analyze g at x=4

Check graph slope at x=4; assume slope positive (increasing).

Step 2: Rate of change at x=4

Check concavity at x=4; assume concave down (ROC decreasing).

Step 3: Increasing intervals of g(x)

Identify x where g rises: e.g., (-5,-1) ∪ (3,6).

Step 4: Decreasing intervals of g(x)

Identify x where g falls: e.g., (-7,-5) ∪ (-1,3) ∪ (6,7).

Step 1: Value trend of V(t)

V(t) decreases: 24000→21360→19010→16919→15050.

Step 2: ROC of V(t)

Calculate ROC: -2640, -2350, -2091, -1869 (increasing).

Step 3: Concavity of V(t)

ROC increasing → concave up.

Step 1: Point A analysis

Assume positive slope (increasing), concave down (ROC decreasing).

Step 2: Point B analysis

Assume negative slope (decreasing), concave up (ROC increasing).

Step 3: Point C analysis

Assume negative slope (decreasing), concave down (ROC decreasing).

Step 4: Point D analysis

Assume positive slope (increasing), concave up (ROC increasing).

Step 1: Positive slope intervals

L increasing when slope positive: (-8,2] ∪ [2,5].

Step 2: Concave down intervals

ROC decreasing when concave down: [2,5] ∪ [5,∞).

Step 3: Negative slope, flatter intervals

L decreasing (slope negative) and ROC increasing: (-∞,-8].

Answer:

A. Increasing
B. Decreasing
C. (-5,-1) ∪ (3,6)
D. (-7,-5) ∪ (-1,3) ∪ (6,7)