1. The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Left Hand Sum Approximation, using the intervals between those given points.
x 10 12 15 19 20
f(x) –2 –5 –9 –12 –16
A. –9.250
B. –10.100
C. –7.550 <———— my choice
D. –6.700
2. The table of values below shows the rate of water consumption in gallons per hour at selected time intervals from t = 0 to t = 12.
Using a right Riemann sum with 5 subintervals estimate the total amount of water consumed in that time interval.
x 0 2 5 7 11 12
f(x) 5.7 5.0 2.0 1.2 0.6 0.4
A.2.742
B. 21.2
C. 32.9
D. None of these
3. The function f is continuous on the interval [3, 13] with selected values of x and f(x) given in the table below. Find the average rate of change of f(x) over the interval [3, 13].
x 3 4 7 10 13
f(x) 2 8 10 12 22
4. f is a differentiable function on the interval [0, 1] and g(x) = f(3x). The table below gives values of f '(x). What is the value of g '(0.1)?
x 0.1 0.2 0.3 0.4 0.5
f '(x) 1 2 3 –4 5
A.1
B. 3
C. 9
D. Cannot be determined
5.
f(x) and g(x) are a differentiable function for all reals and h(x) = g[f(3x)]. The table below gives selected values for f(x), g(x), f '(x), and g '(x). Find the value of h '(1).
x 1 2 3 4 5 6
f(x) 0 3 2 1 2 0
g(x) 1 3 2 6 5 0
f '(x) 3 2 1 4 0 2
g '(x) 1 5 4 3 2 0
Please help me! Oobleck, Thank so you so so much in advance and I greatly appreciate it!(I referred to you because most likely you’re usually the one who responds) so thank you again! I really appreciate all the help :)
#1.
x 10 12 15 19 20
f(x) –2 –5 –9 –12 –16
Approximate the integral by
-2*2 + -5*3 + -9*4 + -12*1 = -67
Care to reconsider?
#2. Just another sum problem.
#3 avg rate of change is just (f(13)-f(3))/(13-3)
#4. You already did one of these. g'(x) = f'(3x)*3
#5. h'(1) = g'(f(3))*f'(1) - Review the chain rule stuff
I don’t understand #2 and #5
Sure, I'll be happy to help you with these questions! Let's go through each one step by step.
1. To estimate the average value of the function using a Left Hand Sum Approximation, we need to calculate the sum of the function values multiplied by the width of each interval, and then divide by the total width of the interval.
First, let's calculate the width of each interval:
Interval 1: 12 - 10 = 2
Interval 2: 15 - 12 = 3
Interval 3: 19 - 15 = 4
Interval 4: 20 - 19 = 1
Now let's calculate the sum of the function values multiplied by the width of each interval:
Sum = (-2 * 2) + (-5 * 3) + (-9 * 4) + (-12 * 1) = -4 - 15 - 36 - 12 = -67
Lastly, let's divide the sum by the total width of the interval:
Average = -67 / (2 + 3 + 4 + 1) = -67 / 10 = -6.7
Therefore, the average value of the function with a Left Hand Sum Approximation is -6.7. The correct answer choice is D.
2. To estimate the total amount of water consumed using a right Riemann sum with 5 subintervals, we need to calculate the sum of the function values multiplied by the width of each interval.
First, let's calculate the width of each interval:
Interval 1: 2 - 0 = 2
Interval 2: 5 - 2 = 3
Interval 3: 7 - 5 = 2
Interval 4: 11 - 7 = 4
Interval 5: 12 - 11 = 1
Now let's calculate the sum of the function values multiplied by the width of each interval:
Sum = (5.0 * 2) + (2.0 * 3) + (1.2 * 2) + (0.6 * 4) + (0.4 * 1) = 10 + 6 + 2.4 + 2.4 + 0.4 = 21.2
Therefore, the estimated total amount of water consumed is 21.2 gallons per hour. The correct answer choice is B.
3. To find the average rate of change of f(x) over the interval [3, 13], we need to calculate the difference in f(x) values divided by the difference in x values.
First, let's calculate the difference in f(x) values for the interval:
Difference in f(x) = 22 - 2 = 20
Now let's calculate the difference in x values for the interval:
Difference in x = 13 - 3 = 10
Finally, let's calculate the average rate of change:
Average rate of change = Difference in f(x) / Difference in x = 20 / 10 = 2
Therefore, the average rate of change of f(x) over the interval [3, 13] is 2.
4. To find the value of g '(0.1), we can use the chain rule. The chain rule states that if g(x) = f(3x), then g '(x) = f '(3x) * 3.
First, let's find the value of f '(0.3) using linear interpolation:
f '(0.2) = 2
f '(0.3) = (3 - 2) / (0.3 - 0.2) = 1 / 0.1 = 10
Now let's calculate the value of g '(0.1):
g '(0.1) = f '(0.3) * 3 = 10 * 3 = 30
Therefore, the value of g '(0.1) is 30. The correct answer choice is D.
5. To find the value of h '(1), we can use the chain rule again. The chain rule states that if h(x) = g[f(3x)], then h '(x) = g '(f(3x)) * f '(3x) * 3.
First, let's find the value of f(3):
f(3) = 2
Now let's find the value of g(2):
g(2) = 2
Now let's find the value of f '(3):
f '(3) = 1
Now let's find the value of g '(2):
g '(2) = 5
Finally, let's calculate the value of h '(1):
h '(1) = g '(f(3)) * f '(3) * 3 = g '(2) * f '(3) * 3 = 5 * 1 * 3 = 15
Therefore, the value of h '(1) is 15.
I hope this helps! Let me know if you have any further questions.