math

Name: MTH 252 (Summer) HW 3

For Homework 3, please explain and show all your steps for all problems. Some of these problems have been slightly modified from Exam 1 problems.

1. Approximate ∫ 6 4

d

dx f(x) dx.

f

t −2 −1 0 1 2 3 4 5 6 7 8

−4 −3 −2 −1

0

1

2

3

4

5

6

7

2. Approximate ∫ 6 3 f(x) dx.

Math 252 (Summer) HW 3 Page 2 of 5

3. The population of stray cats in Integral City has been growing at a rate of R(t) = 3t2 + 3 cats per year since 1981, where t is the number of years since 1981 (i.e t = 10 is ten years after 1981, which is 1991). In 1982, an animal surveyor estimated 10 stray cats lived in Integral City. Write the function s(t) that represents the number of cats t years since 1981 (given the information provided).

4. A roller coaster has a height (in feet) modeled by the function f(t) = 6t3 − 4t2 + 50, where time t is measured in minutes. What is the average height between 0 and 2 minutes?

Math 252 (Summer) HW 3 Page 3 of 5

5. Write the limit of Right Riemann sums below as a definite integral. Do not evaluate.

lim n→∞

n∑ i=1

   [−2 + 5

n i

]2 + 2

[ −2 +

5

n i

] 5 n

  =

6. A(x) = ∫ −2 x

e3tdt. Calculate A′(x).

Math 252 (Summer) HW 3 Page 4 of 5

For problems 7 - 8, use the table below.

x −2 0 2 4 6 8 10 f(x) 2 0 1 −1 3 2 −1

7. Plot the points from the above table, and draw in the rectangles associated with a Left Riemann Sum and 3 subintervals.

f

x −2 0 2 4 6 8 10

−3 −2 −1

0

1

2

3

4

8. Now estimate ∫ 10 −2

f(x)dx using a Left Riemann Sum with 3 subintervals

Math 252 (Summer) HW 3 Page 5 of 5

9. Suppose f is a continuous function with f(2) = 1, and f(8) = −2. Let g(x) =

∫ x3 −1

f(t)dt. Evaluate g′(2).

10. Integrate ∫ x √

5 −x dx