6) on the interval 6,9, the graph on the right is.. (select all that apply)\na) the function values are…

6) on the interval 6,9, the graph on the right is.. (select all that apply)\na) the function values are increasing\nb) the function values are decreasing\nc) the roc is increasing\nd) the roc is decreasing\ne) the roc is positive\nf) the roc is negative\ng) the change in roc is positive\nh) the change in roc is negative\n7) determine if the function in each table below is concave up, down, or neither. justify your answer.\n| x | -12 | -11 | -10 | -9 | -8 | -7 |\n|----|----|----|----|----|----|----|\n| f(x) | -15 | -11 | -8 | -6 | -5 | -5 |\n| x | -17 | -10 | -5 | -2 |\n|----|----|----|----|----|\n| g(x) | 50 | 36 | 26 | 20 |\n8) over which interval is the total change of the function g(x) least?\n| interval | 0,4 | 4,5 | 5,8 | 8,10 |\n|----|----|----|----|----|\n| average roc | 1 | -7 | 3 | -4 |\n9) f(x) is increasing on (0,3) ∪ (7,10) and decreasing on (3,7). g(x) is concave down, with a maximum at x = 5. on which interval in (0,10) is the function h(x)=f(x)+g(x) guaranteed to be decreasing?\n10) suppose k(x)=0 and k(x)=h(x + 1)-h(x). assume h(x) is negative. circle the best choice below. if the best choice is neither, do not circle a choice.\nthen h(x) is zero / constant / linear / quad\nalso suppose h(x)=f(x + 1)-f(x). circle the best choice below.\nthen f(x) is zero / constant / linear / quad, and increasing / decreasing, and concave up / down\nfinally suppose f(x)=g(x + 1)-g(x). circle the best choice below.\nthen g(x) is zero / constant / linear / quad, and increasing / decreasing, and concave up / down

6) on the interval 6,9, the graph on the right is.. (select all that apply)\na) the function values are increasing\nb) the function values are decreasing\nc) the roc is increasing\nd) the roc is decreasing\ne) the roc is positive\nf) the roc is negative\ng) the change in roc is positive\nh) the change in roc is negative\n7) determine if the function in each table below is concave up, down, or neither. justify your answer.\n| x | -12 | -11 | -10 | -9 | -8 | -7 |\n|----|----|----|----|----|----|----|\n| f(x) | -15 | -11 | -8 | -6 | -5 | -5 |\n| x | -17 | -10 | -5 | -2 |\n|----|----|----|----|----|\n| g(x) | 50 | 36 | 26 | 20 |\n8) over which interval is the total change of the function g(x) least?\n| interval | 0,4 | 4,5 | 5,8 | 8,10 |\n|----|----|----|----|----|\n| average roc | 1 | -7 | 3 | -4 |\n9) f(x) is increasing on (0,3) ∪ (7,10) and decreasing on (3,7). g(x) is concave down, with a maximum at x = 5. on which interval in (0,10) is the function h(x)=f(x)+g(x) guaranteed to be decreasing?\n10) suppose k(x)=0 and k(x)=h(x + 1)-h(x). assume h(x) is negative. circle the best choice below. if the best choice is neither, do not circle a choice.\nthen h(x) is zero / constant / linear / quad\nalso suppose h(x)=f(x + 1)-f(x). circle the best choice below.\nthen f(x) is zero / constant / linear / quad, and increasing / decreasing, and concave up / down\nfinally suppose f(x)=g(x + 1)-g(x). circle the best choice below.\nthen g(x) is zero / constant / linear / quad, and increasing / decreasing, and concave up / down

Answer

6)

Explanation:

Step1: Analyze function - value trend

On the interval ([6,9]), as (x) increases, the (y -)values (function values) of the graph are going up. So the function values are increasing, and option A is correct.

Step2: Analyze rate - of - change (ROC)

The rate of change (ROC) of a function (y = f(x)) on an interval ([a,b]) is given by (\frac{f(b)-f(a)}{b - a}). Since the function is increasing on ([6,9]), the ROC is positive. So option E is correct.

Step3: Analyze change in ROC

The graph is getting less steep on ([6,9]), which means the rate of change is decreasing. So option D is correct.

Answer:

A. The function values are increasing, D. The ROC is decreasing, E. The ROC is positive

7)

For the first table:

Explanation:

Step1: Calculate first - differences

The first - differences of (y = f(x)) are: (f(-11)-f(-12)=-11 + 15 = 4), (f(-10)-f(-11)=-8 + 11 = 3), (f(-9)-f(-10)=-6 + 8 = 2), (f(-8)-f(-9)=-5+6 = 1), (f(-7)-f(-8)=-5 + 5 = 0).

Step2: Analyze concavity

Since the first - differences are decreasing, the function (y = f(x)) is concave down. For the second table:

Step3: Calculate first - differences

The first - differences of (y = g(x)) are: (g(-10)-g(-17)=36 - 50=-14), (g(-5)-g(-10)=26 - 36=-10), (g(-2)-g(-5)=20 - 26=-6).

Step4: Analyze concavity

Since the first - differences are increasing (the absolute values are decreasing), the function (y = g(x)) is concave up.

Answer:

The function (f(x)) is concave down. The function (g(x)) is concave up.

8)

Explanation:

Step1: Recall the formula for total change

The total change of a function (y = g(x)) on an interval ([a,b]) is (g(b)-g(a)). The average rate of change (ROC) on ([a,b]) is (\frac{g(b)-g(a)}{b - a}). The total change is the product of the average ROC and the length of the interval.

Step2: Calculate total change for each interval

For the interval ([0,4]): Let the average ROC be (m_1 = 1) and the length of the interval (\Delta x_1=4-0 = 4), so the total change (C_1=m_1\Delta x_1=1\times4 = 4). For the interval ([4,5]): Let the average ROC be (m_2=-7) and the length of the interval (\Delta x_2=5 - 4 = 1), so the total change (C_2=m_2\Delta x_2=-7\times1=-7). For the interval ([5,8]): Let the average ROC be (m_3 = 3) and the length of the interval (\Delta x_3=8 - 5 = 3), so the total change (C_3=m_3\Delta x_3=3\times3 = 9). For the interval ([8,10]): Let the average ROC be (m_4=-4) and the length of the interval (\Delta x_4=10 - 8 = 2), so the total change (C_4=m_4\Delta x_4=-4\times2=-8). The smallest absolute - value of the total change is for the interval ([4,5]).

Answer:

([4,5])

9)

Explanation:

Step1: Recall the sum - of - functions property

The derivative of (h(x)=f(x)+g(x)) is (h^\prime(x)=f^\prime(x)+g^\prime(x)). The function (h(x)) is decreasing when (h^\prime(x)<0).

Step2: Analyze the derivatives of (f(x)) and (g(x))

(f(x)) is decreasing on ((3,7)) so (f^\prime(x)<0) on ((3,7)), and (g(x)) is concave down with a maximum at (x = 5), so (g^\prime(x)<0) for (x>5).

Step3: Determine the interval for (h(x))

To have (h^\prime(x)=f^\prime(x)+g^\prime(x)<0), we consider the intersection of the intervals where (f^\prime(x)<0) and (g^\prime(x)<0). The intersection of ((3,7)) and ((5,10)) is ((5,7)).

Answer:

((5,7))

10)

  1. If (k(x)=h(x + 1)-h(x)=0) for all (x), then (h(x + 1)=h(x)) for all (x). So (h(x)) is constant.
  2. If (h(x)=f(x + 1)-f(x)) and (h(x)) is negative, then (f(x+1)-f(x)<0) for all (x), so (f(x)) is decreasing. Also, since (h(x)) is constant (because (h(x)) has a constant non - zero value), (f(x)) is linear. And the second - derivative of (f(x)) is (0) (because the first - derivative (h(x)) is constant), so (f(x)) is neither concave up nor concave down.
  3. If (f(x)=g(x + 1)-g(x)), without more information, we cannot determine the nature of (g(x)) precisely. But if (f(x)) is constant, then (g(x)) is linear. If (f(x)) is non - constant, we need more data. Assuming (f(x)) is constant, (g(x)) is linear. Since we don't know if (f(x)) is positive or negative, we can't determine if (g(x)) is increasing or decreasing, and since (g(x)) is linear, it is neither concave up nor concave down.

Answer:

  1. (h(x)) is constant.
  2. (f(x)) is linear, decreasing, neither concave up nor concave down.
  3. Insufficient information to fully determine (g(x)) (if we assume (f(x)) is constant, (g(x)) is linear, and we can't determine increasing/decreasing and concavity without more info about (f(x)) sign).