I need help solving these questions and figuring out the solutions

 

 

1. Find each requested anti-derivative.
a) ∫ x dx
4 b) ∫ 3 c) ∫ e6 x dx x 2 dx d) 5 ∫ x dx 2. Find the specific function f(x) which satisfies the following information:
a) f ′(=
x) 3x 2 − 8 x and b) f ′=
( x) and 6x − 2 f( 2 ) = 3
f( 3 ) = 1 (Hint: use substitution) 3. Use the Method of Substitution to find the values of each Integral
b) ∫ x e5 x a) ∫ 4 x 2 ( x3 − 1)6 dx 2 +7 dx 4. Solve the differential equation. Find the function satisfying the given information.
a) dy
=
4 x − 6 and
dx y (2) =
1 dy
b) = e 2 x + 3
dx 1
1 2 2
( x − 3) 2 + 4=
x − x + 5 on the interval [0,12]
9
9
3
using n = 4 rectangles and Right-hand endpoints. (Hint: a sketch might help) 12pts 5. Estimate the Area under the curve of f ( x)= 6. Given the following information regarding f(x), answer the questions below.
10 4 7 1 10 1 14 , ∫ f ( x)dx =
6
−3 , ∫ f ( x)dx =
∫ f ( x)dx =
4 a) ∫ 10 f ( x)dx 1 b) ∫ 2 f ( x)dx
4 7 c) ∫ f ( x)dx
4 7. Find the value of each Definite Integral, NO DECIMALS
16 a) ∫ 5 b) xdx 4 − 6 x 2 + 7)dx −3 9 10 3
c) ∫ dx
x
2 ∫ (x 2 d) ∫ e dx
4x −1 x2 − 4 x
∫ x 4 dx
−2
1 e) 8. Use the Substitution Method to evaluate the Definite Integrals below
2 a) 6 2
5
∫ x( x − 1) dx b) ∫x 10 − x dx −6 0 9. Find the following anti-derivatives.
a) ∫ (10 x 4 + 2
)dx
x5 3x
∫ xe dx
2 b) 10. Find the specific function f(x) which satisfies all of the following:
1
a) f ′( x) =
and f (11) 2
4x + 5
b) f ′( x) = x3 − 4 x and f (−6) = 4 11. Evaluate each of the Definite Integrals. NO DECIMAL ANSWERS.
2 2 a) ∫ (2 x 3 b) ∫ 36 x 2 ( x3 + 1)5 dx − 3 x )dx
2 −2 0 9 5 13 2 9 5 12. Given=
∫ f ( x)dx 10,=
∫ f ( x)dx 7,=
∫ f ( x)dx 16 , find
9 13 a) ∫ f ( x)dx b) 2 ∫ 5 f ( x)dx 13 13. Find the Area bounded by =
y x 2 and =
y 2x + 8
3
14. Find the Area bounded
by y x=
=
and y 9 x 15. Ed starts a 401k account at age 27, putting away $6000 each year. If the account earns 7.2%
continuously, how much will be in the account when Ed reaches age: (i) 60 (ii) 65 (iii) 70
16. An investment opportunity is presented, which will provide a continuous income stream worth
$75000 each year for the next 8 years. Assuming an earnings rate of 5% continuously, what is this worth
in today’s dollars?
17. Marie has built up a retirement account with a total value of $1,250,000 and wishes to convert it into
an annuity that will pay her out over the next 25 years. Assuming an earnings rate of 6.4% continuously,
how much will this provide her each year? (Hint: use the PV formula) 0 ≤ x ≤ 10
elsewhere 1/10
18. A PDF is given: f ( x) = 0 a) Show that this satisfies Principles I and II of a Probability Density Function
b) Use Principle III to find P( 2 < X < 8) 3(1 − x 2 ) 19. A PDF is given: f ( x) = 4 0 −1 ≤ x ≤ 1
elsewhere a) Show that this satisfies Principles I and II of a Probability Density Function
b) Use Principle III to find P( -1/2 < X < 1/4 ) 20. Find each Improper Integral’s value, if is Converges, or show why it Diverges
∞ a) 3
∫2 x5 dx ∞ b) ∫
1 3
dx
3
x ∞ ∞ c) 3
∫2 x dx d) 1 ∫2e
1 − t /2 dt

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• 1492676420-Math136 Midterm2 practice 1.pdf