MATH 1914 Homework 10
Section 3.9 Problems
1. Find the most general antiderivative of the function. (Check your answer by differentiation)
f (x) = 7x + 4
2. Find the most general antiderivative of the function. (Check your answer by differentiation)
7
f (x) = 3 + 5 cos(x) + √
3
x
3. Find f .
f 00 (x) = 18x3 − 14x2 + 16x + 2
4. Find f .
f 000 (x) = sin(x) f (0) = 2 f 0 (0) = 1 f 00 (0) = 3
5. The graph of a function is shown in the figure. Make a rough sketch of an antiderivative
F , given that F (0) = 1.
1
6. The graph of the velocity function of a particle is shown in the figure. Sketch the
graph of a position function.
7. The graph of f 0 is shown in the figure. Sketch the graph of f is f is continuous and
f (0) = −1.
Section 4.1 Problems
8. (a) Use six rectangles to find estimates of each type for the area under the given graph of
f from x = 0 to x = 12
(i) L6 (sample points are left endpoints)
(ii) R6 (sample points are right endpoints)
(iii) M6 (sample points are midpoints)
(b) Is L6 an underestimate or an overestimate of the true area?
(c) Is R6 an underestimate or an overestimate of the true area?
(d) Which of the numbers L6 , R6 , or M6 gives the best estimate? Explain.
9. (a) Estimate the area under the graph of f (x) = sin(x) from x = 0 to x =
2
π
2
using
four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is
your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints.
10. Evaluate the upper and lower sums for f (x) = 2 + x2 , −1 ≤ x ≤ 1 with n = 3
and 4. Illustrate with diagrams like Figure 14 in Section 4.1.
11. The velocity graph of a car accelerating from rest to a speed of 120 km/h over a
period of 30 seconds is shown. Estimate the distance traveled during the period.
12. Use Definition 2 in Section 4.1 to find an expression for the area under the graph
of f as a limit. Do not evaluate the limit.
√
f (x) = x2 + 1 + 2x 4 ≤ x ≤ 7
13. Determine a region whose area is equal to the given limit. Do not evaluate the limit.
n
X
π
iπ
lim
tan
n→∞
2n
4n
i=1
3
Section 4.2 Problems
14. Evaluate the Riemann sum for f (x) = x − 3, −6 ≤ x ≤ 4, with five subintervals, taking
the sample points to be right endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.
R 10
15. The graph of a function f is given. Estimate 0 f (x)dx using five subintervals with (a)
right endpoints, (b) left endpoints, and (c) midpoints.
16. A table of values ofR an increasing function f is shown. Use the table to find lower
30
and upper estimates for 10 f (x)dx.
x
10 14 18 22 26 30
f(x) -12 -6 -2 1 3 8
17. Use the Midpoint Rule with the given value of n to approximate the integral. Round
the answer to four decimal places.
Z 1√
x3 + 1dx, n=3
0
18. Express the limit as a definite integral on the given interval.
lim
n→∞
n
X
xi
q
1 + x3i ∆x, [2, 5]
i=1
19. RThe graph of f is shown. Evaluate each integral by interpreting it in terms of area.
2
(a) 0 f (x)dx
R5
(b) 0 f (x)dx
4
R7
(c) 5 f (x)dx
R9
(d) 0 f (x)dx
20.Evaluate the integral by interpreting it in terms of areas.
Z 4
√
(x − 16 − x2 )dx
−4
21. Use Property 8 in Section 4.2 to estimate the value of the integral.
Z 3
1
dx
0 x+4
5