kim (kdk738) HW 7 yin (54960)
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Find the absolute maximum of
f (x) = 3 + 4(4 x)ex3
on the int
Problem Set 19
1. Find the volume of solid whose base is the
given region and whose cross section perpendicular to the X - axis is a square.
(a) The triangular region bounded by coordinate axes and the line y = 3 x.
2. Find the volume of solid obtained by
Problem Set 16
1. Verify by dierentiation that the formula
sin + sin tan2
cos x dx = sin x sin3 x + C
x( 3 x + 4 x) dx
dx = 2 (bx2a) a + bx+C
a + bx
(x 2|x|) dx
2. Evaluate the in
Problem Set 15
1. Evaluate the integral by interpreting in
terms of areas.
1 |x| dx
3 (1 + 9
x2 ) dx
2. Use properties of integrals to verify the inequality without evaluating the integral.
(a) 0 1 + x2 dx 0 1 +
Problem Set 14
1. Express the area under the graph of f as
a limit. Do not evaluate the limit.
(a) f (x) = 4 x, 1 x 16
5. Use midpoint rule with the given value of
n to approximate the integral. Round the
answer to 4 decimal places.
x + 1 dx, n = 4
Problem Set 20
Find the Volume of the described solid S:
A cap of a sphere with radius r and height h
A right circular cone with height h and radius r
The base of S is an elliptical region with boundary curve:
. Cross-sections perpendicular
Problem Set 13
6. Determine whether the statement is true
or false. If it is true, explain why. If it is
false, explain why or give an example that
disproves the statement.
1. Find the most general antiderivative of the
function. (Check your answer by die
Problem Set 11
If f changes from negative to positive at a critical number c, then c is
a local minimum. Second derivative
If f (c) = 0 and f (c) = 0 then,
f (c) > 0 local minimum
f (c) < 0 local maximum.
(g) Concavity and points of inection :
Problem Set 12
9. A cone with height h is inscribed in a
larger cone with height H so that its vertex is at the center of the base of the larger
cone. Show that the inner cone has maximum volume when h = 1 H .
1. Show that of all rectangles with a given
Problem Set 9
Where s is in meters and t is in seconds.
Find the largest and smallest values of its
velocity during 1 t 5.
7. Show that | sin x cos x| 2 for x
[0, 2 ] and hence, for all x.
1. Find critical points of each function.
(a) f (x) = 4x3 5x2 8x
Problem Set 10
6. Two runners start the race at the same
time and nish in a tie. Prove that at
some time during the race they have the
1. Verify if the function f satises the hypothesis of MVT on the given interval
[a, b]. Then, nd all numbers
Problem Set 8
3. Find the indicated rate.
1. A particle moves according to the law of
motion, s = f (t) = cos( t ), where s is in
meters and t in seconds.
(a) If 5x2 y = 100 and
when x = 10
(b) If y = 2 x 9 and
when x = 9
(a) Find the velocity at time t
Problem Set 5
1. For the function f (x) =
x + 1,
(a) Find the rst derivative f (x).
(b) Determine the interval over which
f (x) exists.
y = f(x)
(c) Find the equation of tangent line at
x = 3.
2. For the function f (x) = x3 + x,
(a) Find the
Problem Set 18
1. Sketch the region bounded by the given
curves. Decide whether to integrate w.r.t.
x or y. Then nd the area of the region
(a) y = x2 , y = x, x = 1, x = 1
(b) y = sin x. y = x, x = /2, x =
(c) x = 2y 2 , x = 4 + y 2
(d) y = 12
Problem Set 17
1. Evaluate the integral by making the given
x3 (2 + x4 )5 dx, u = 2 + x4
cos3 sin d, u = cos
, u = 1/x
2. Given below are pairs of integration problems. One of them requires