Animals - Calculus - Exam, Exams for Calculus. Punjab Engineering College

Calculus

Description: Key points of this exam are: Animals, Multiplying Constant, Function, Present, Integrand, Decreasing, Concave Up, Quantities, Ascending Order, Rate of Change
Showing pages  1  -  4  of  4
Math 106: Review for Exam I
1. Find the following. [Substitution tip: usually let u= a function that’s “inside” another function,
especially if du (possibly oﬀ by a multiplying constant) is also present in the integrand.]
(a) Z4
1
ex
xdx
(b) Z2π
π
cos7(5x) sin(5x)dx
(c) Z7x2
1 + x6dx
(d) Z10
6
x10 x dx
2. Suppose f(x) is decreasing and concave up.
(a) Put the following quantities in ascending order.
L100,R100,T100,M100,Zb
a
f(x)dx
(b) What can you say with certainty about where S200 would ﬁt into your list above?
3. Suppose f(t) is the rate of change (in animals per month) of a population P(t).
(a) What does Z12
4
f(t)dt represent in this problem?
(b) Find the best possible left, right, midpoint, trapezoidal, and Simpson’s approximations to Z12
4
f(t)dt
given the data in the table below.
t4 6 8 10 12
f(t) 15 11 8 4 3
4. Find bounds for each of the following errors if I=Z7
2
ln x dx.
(a) |IL100|
(b) |IT100|
(c) |IM100|
5. If I=Z7
2
ln x dx, how many subdivisions are required to obtain a trapezoidal sum approximation with
error of at most 1/1,000,000?
6. Use Euler’s method with three steps on the diﬀerential equation dy
dt =ytto estimate y(2.5) if
y(1) = 0.
7. Solve the diﬀerential equation dy/dx = 2xy + 6xif the solution passes through (0,5). [Students in the
8:00 section should omit this problem.]
8. Write integrals equal to
(a) the arc length of y=x2on the interval [1,5]
(b) the area bounded by y=x28x+ 24 and y= 3x
9. Consider the region bounded by y=x,y= 0, and x= 9. Write an integral equal to the volume
generated if this region is revolved about
(a) the x-axis
(b) the line x=1
10. A pyramid has a square base 30 feet to a side and a height of 10 feet. Write integrals equal to
(a) the volume of the pyramid
(b) the work done in pumping all the ﬂuid to a point 5 feet above the pyramid if the pyramid is ﬁlled
to a height of 8 feet with water (which weighs 62.4 pounds per cubic foot)