My question is problem #11 and 1.a) on this assignment

 

 

3rd ASSIGNMENT
DUE DATE: FRIDAY, JUNE 24TH 1. Differential Equations and Exponential
Growth/Decay
Problem 1. The isotope thorium-234 has a half-life of 24.5 days.
(a) What is the di↵erential equation satisfied by y(t), the amount of
thorium-234 in a sample at time t?
(b) At t = 0, a sample contains 2 kg of thorium-234. How much
remains after 40 days?
Problem 2. Verify that if the function f (x) is a solution to the corresponding di↵erential equation:
(a) f (x) = 4e3x sin x, y 0 = 3y + 4e3x cos x
(b) f (x) = e3x , y 00 + 2y 0 · 15y = 0
(c) f (x) = ex , y 00 = x
2
2
(d) f (x) = 2x2x 3 , y 0 = y (xx3 3)
2. Integration by Substitution
Problem 3. Compute the following indefinite integrals, using the suggested substitution.
R
R
(a) sec2 x tan x dx, u = tan x
(d) R sin2 ✓ cos ✓ d✓, u = sin ✓
R
2
(e) R sin(4✓
7)d✓, u = 4✓ 7
(b) xe x dx, u = x2
p
2
(arctan x)2
(f) x x + 1 dx, u = x + 1
(c)
dx, u = arctan x
2
x +1 Problem 4. Compute the following indefinite integrals:
R
R p
(a) R (x2 + 1)(x3 + 3x)4 dx
(f) R x3 px2 + 1 dx
(b) cot x csc 2x dx
(g) x2 2 + x dx
R cos(⇡/x)
R (ln u)4
(c)
dx
(h)
2
x
R udx du
R sin px
p
p
(i)
(d)
dx
R 1x x
R (1+1/5x)4
(e) xe dx
(j) x
tan(x4/5 )dx
1 DUE DATE: FRIDAY, JUNE 24TH 2 Problem 5. Solve the following di↵erential equations with given initial
value:
(a) y 0 =
(b) y 0 =
(c) y 0 = cos x
, y( ⇡2 ) = 3
sin2 x
p2x , y(0) = 0
1 x
dx
p
, y( ⇡4 )
cos2 x 1+tanx cos x
⇡
(d) y 0 = sin
)=0
2 x dx, y(
4
p
0
x
x
(e) y = e 1 e , y( ln 2) = 0
(f) y 0 = x(2x + 5)8 , y(1) = 2 =7 Problem 6. Can They Both Be Right? HannahR uses the substitution
u = tan x and Akiva uses u = sec x to evaluate tan x sec2 xdx. Show
that they obtain di↵erent answers, and explain the apparent contradiction.
R
Problem 7. Evaluate sin x cos x dx using substitution in two di↵erent ways: first using u = sin x and then using u = cos x. Reconcile the
two di↵erent answers.
Problem 8. Some ChoicesZ Are Better Than Others. Evaluate
sin x cos2 x dx
twice. First use u Z= sin x to show that
Z p
2
sin x cos x dx = u 1 u2 du and evaluate the integral on the right by a further substitution. Then
show that u = cos x is a better choice.
3. Fundamental Theorem of Calculus, part II
Problem 10. Find the smallest positive critical point of
Z x
F (x) =
cos(t3/2 )dt
0 and determine whether it is a local min or max.
Problem 11. Evaluate
d
dx Z ex sin t dt .
ln x 4. Area Between Curves
Problem 12.
(a) Find the area of the region between y = 3x2 + 12 and y =
4x + 4 over [ 3, 3]: 3rd ASSIGNMENT 3 (b) Find the area of the region enclosed by the graphs of f (x) =
x2 + 2 and g(x) = 2x + 5: Problem 13. Sketch the region bounded by
1
1
y=p
and y = p
1 x2
1 x2
for 12 x 12 and find its area.
Problem 14. Sketch the region bounded by the curves and compute
its area:
(a) y = x + 1 , y = 9 x2 , x = 1 and x = 2
(b) y = x12 , y = x and x = 18 x
(c) y = 3x2 , y = 8x2 , 4x + y = 4 and x 0
5. The Method of Partial Fractions
Problem 15. Write out the form of the partial fraction decomposition
of the function:
(a)
(b) 2x
(x+3)(3x+1)
1
x2 1 (c)
(d) 4x2 7x 12
x(x+2)(x 3)
x2 +2x 1
x3 x

Attachments:

• 1492851607-MATH 1001 assignment 3.pdf