since you want 4 subintervals to cover the distance, each will have width=30. So,
∫[0,120] f(x) ≈
4
∑ 30f(xk)
k=1
where xk is the left, middle, or right value for the subinterval.
The trapezoidal rule, or course, averages the left and right values of f(xk), since those are the bases of the trapezoids.
For the graph,
clearly the domain of f(t) = h'(x) is [0,6], since f is continuous there.
But the domain of h(x) is not the same, since for 0 < t < 1/2, 2t-1 < 0, so f(2t-1) is undefined there. Similarly, for t>7/2, 2t-1 > 6, so f is undefined there.
So, the domain of h(x) is [1/2, 7/2]
at t=5/2, f(t) is a circle with radius 2, with center at (2,0). So,
f^2(5/2) + (1/2)^2 = 2^2
f(5/2) = √15/2 = h'(5/2)
This one is kinda tricky, since h(x) is not just ∫[0,x] f(t) dt
To see the graph of h(x), go to desmos.com (a very handy tool!) and enter two lines of text:
f(t) = {0<=t<=4: sqrt(4-(t-2)^2), 4<t<=6: sqrt(1-(t-5)^2)}
h(x) = int0^2x-1v f(t) dt
it will do typesetting for you, and the ^ in the integral means hit the up-arrow so it changes from the lower limit to the upper limit, and the "v" indicates the down-arrow, so it drops down to the line for the integrand.
h has a max when h' = f = 0, and h" = f' < 0
That is, at t=4
Hello! I am trying to create something to study from, so if someone could give the correct answers and explanations I would be very grateful!
1. Measurements of a lake’s width were taken at 15-foot intervals, as shown:
x: 0, 15, 30, 45, 60, 75, 90, 105, 120
f(x): 0, 15, 18, 20, 19, 23, 24, 22, 12
Estimate (120 on top, 0 on bottom) ∫f(x)dx with n = 4, using each of the following methods.
A. Left-hand approximation
B. Right-hand approximation
C. Trapezoidal approximation
D. Midpoint
2. Let f be the function shown below, with domain the closed interval [0, 6]. The graph of f shown below is composed of two semicircles.
(graph) gyazo.com/3520e36ca7d1770b2690116ca742e7e5
A. Determine the domain of h(x).
B. Find h'(5/2)
C. At what x is h(x) a maximum? Show all the analysis that leads to your conclusion.
2 answers
Thank you so much for the help!
Do you think you could expand a bit on the Left-hand approximation Right-hand approximation, Trapezoidal approximation, and Midpoint?
Also, is there any way of doing part 2C without graphing? Again, thank you so much for using up your time and effort to help me!
Do you think you could expand a bit on the Left-hand approximation Right-hand approximation, Trapezoidal approximation, and Midpoint?
Also, is there any way of doing part 2C without graphing? Again, thank you so much for using up your time and effort to help me!
2 answers