Question 1

Evaluate $\int _ { 2 } ^ { 10 } \frac { 1 } { ( 2 x + 5 ) \sqrt { ( 2 x + 5 ) } } d x$
A)2,882
B)766
C)625
D)525.8
E)576.4

Accepted Answer

To solve the integral $\int _ { 2 } ^ { 10 } \frac { 1 } { ( 2 x + 5 ) \sqrt { ( 2 x + 5 ) } } d x$, we first simplify the integrand. Notice that $\frac { 1 } { ( 2 x + 5 ) \sqrt { ( 2 x + 5 ) } }$ can be rewritten as $(2x + 5)^{-3/2}$. The integral becomes $\int _ { 2 } ^ { 10 } (2x + 5)^{-3/2} dx$.To integrate, let $u = 2x + 5$, thus $du = 2dx$, and $dx = \frac{du}{2}$. When $x = 2$, $u = 9$, and when $x = 10$, $u = 25$. Substituting these into the integral gives us $\frac{1}{2}\int _ { 9 } ^ { 25 } u^{-3/2} du$.The antiderivative of $u^{-3/2}$ is $-2u^{-1/2}$, so the integral becomes $-\left[\frac{u^{-1/2}}{2}\right]_{9}^{25}$ = $-\left[\frac{1}{\sqrt{u}}\right]_{9}^{25}$.Evaluating this from 9 to 25 gives $-\left(\frac{1}{5} - \frac{1}{3}\right) = \frac{1}{3} - \frac{1}{5} = \frac{5-3}{15} = \frac{2}{15}$.However, it seems there was a mistake in the calculation or interpretation of the options provided. The correct evaluation process does not match any of the given options exactly, suggesting a possible error in the setup or in the options themselves. The integral, when evaluated correctly, should yield a specific numerical value, but none of the options provided (A through E) accurately reflect the correct process or outcome for this integral calculation.

Question 2

If the investment flow is $I ( t ) = 5000 t ^ { \frac { 1 } { 4 } }$ calculate the capital formation from the end of the second year to half- way through the fifth year. Give your answer to the nearest whole number.
A)33,691
B)20,879
C)16,703
D)26,217
E)24,177

Accepted Answer

C

Question 3

Which of the following is an expression for g'(x)when
$g ( x ) = \int _ { 2 } ^ { x } \left( 5 t ^ { 2 } - 2 t \right) d t ?$
A) $5 t ^ { 2 } - 2 t$
B) $5 x ^ { 2 } - 2 x$
C) $\frac { 5 } { 3 } x ^ { 3 } - \frac { 2 } { 3 } x ^ { 2 }$
D) 0
E) $\frac { 5 } { 3 } t ^ { 3 } - \frac { 2 } { 3 } t ^ { 2 }$

Accepted Answer

Taking the derivative of the integral function results in the integrand function, so $g'(x) = 5x^2-2x.$ Therefore, the best choice is B.

Question 4

Find $\int x ^ { 3 } - \frac { 5 } { x ^ { 6 } } + \frac { 2 } { x } - 4 e ^ { - 4 x } d x$
A) $\frac { x ^ { 9 } + 4 } { 4 x ^ { 5 } } + \ln x ^ { 2 } + e ^ { - 4 x } + c$
B) $\frac { x ^ { 9 } + 4 } { 4 x ^ { 5 } } + 2 \ln x ^ { 2 } - e ^ { - 4 x } + c$
C) $\frac { x ^ { 9 } - 4 } { 4 x ^ { 5 } } + \ln x ^ { 2 } - e ^ { - 4 x } + c$
D) $\frac { x ^ { 9 } + 4 } { 4 x ^ { 5 } } + 2 \ln x - e ^ { - 4 x } + c$
E) $\frac { x ^ { 9 } + 4 } { 4 x ^ { 5 } } + 0.5 \ln x + e ^ { - 4 x } + c$

Accepted Answer

The answer of Find \(\int x ^ { 3 }...

Question 5

Which of the following represents the consumption function when
$M P C = 0.4 + \frac { 0.1 } { \sqrt { Y } }$, and $C = 50$ when $Y = 100$ ?
A) $0.4 Y + 0.2 \sqrt { Y } + 8$
B) $0.4 Y + 0.2 \sqrt { Y } + 50$
C) $0.4 Y + 0.05 \sqrt { Y } + 50$
D) $0.4 Y - 0.2 \sqrt { Y } + 50$
E) $0.4 Y - 0.05 \sqrt { Y } + 5$

Accepted Answer

Given $MPC = 0.4 + \frac{0.1}{\sqrt{Y}}$, the consumption function can be derived by integrating the MPC with respect to income (Y), since the marginal propensity to consume (MPC) is the derivative of the consumption function with respect to income. Given $C = 50$ when $Y = 100$, we can solve for the constant of integration. The correct formula that fits the given conditions is $0.4Y + 0.2\sqrt{Y} + 8$.

Question 6

Find the consumer's surplus at Q = 8 for the demand function P = 50 - 4Q.
A)272
B)256
C)298
D)192
E)128

Accepted Answer

The answer of Find the consumer's surplus at Q =...

Question 7

The producer's surplus at Q = a for the supply function P = 6 + 8Q
Is known to be 400. Find the value of a.
A)25
B)5
C)20
D)10
E)15

Accepted Answer

The producer's surplus is the area above the supply curve and below the price level at Q = a. The supply function is P = 6 + 8Q. At Q = a, the price is P = 6 + 8a. The producer's surplus is given by the integral from 0 to a of (P - supply price) dQ, which equals 400. Solving the equation 400 = ∫(0 to a) (6 + 8a - (6 + 8Q)) dQ gives a = 10.

Question 8

Evaluate $\int _ { - 1 } ^ { 2 } 5 x ^ { 2 } - 4 x + 6 d x$
A) $11 \frac { 2 } { 3 }$

B) 27

C) $23 \frac { 2 } { 3 }$

D) $15 \frac { 2 } { 3 }$

E) 21

Accepted Answer

The answer of Evaluate \(\int _ { - 1 }...

Question 9

Which of the following expressions represents the present value of a continuous revenue stream for n years at a constant rate of $S per year if the discount rate is r%?
A) $\frac { r } { 100 S } \left[ e ^ { \frac { - m } { 100 } } + 1 \right]$

B) $\frac { r S e ^ { \frac { - m } { 100 } } } { 100 }$

C) $\frac { \boldsymbol { r } } { 100 S } \left[ e ^ { \frac { - \boldsymbol { m } } { 100 } } - 1 \right]$

D) $\frac { 100 S e ^ { \frac { - m } { 100 } } } { r }$

E) $\frac { 100 S } { r } \left[ 1 - e ^ { \frac { - m } { 100 } } \right]$

Accepted Answer

The present value of a continuous revenue stream is given by the formula PV = (C/r)(1-e^(-rt)), where C is the constant revenue stream, r is the discount rate and t is time. In this case, since the revenue stream is for n years, time will be n. Therefore, the expression that represents the present value is PV = (S/r)(1-e^(-rn)). Simplifying this expression, we get PV = (100S/r)(1-e^(-(r/100)n)). This expression matches option E, which is the correct answer.

Question 10

A firm's marginal revenue and marginal cost functions are given by 
MR = 240 - 0.6Q2 and MC = 150 + 0.3Q2
If fixed costs are 50, determine the maximum profit.
A)200
B)550
C)800
D)650
E)150

Accepted Answer

The answer of A firm's marginal revenue and marginal cost...

Quiz 6: Integration

Evaluate $\int _ { 2 } ^ { 10 } \frac { 1 } { ( 2 x + 5 ) \sqrt { ( 2 x + 5 ) } } d x$

If the investment flow is $I ( t ) = 5000 t ^ { \frac { 1 } { 4 } }$ calculate the capital formation from the end of the second year to half- way through the fifth year. Give your answer to the nearest whole number.

Which of the following is an expression for g'(x) when $g ( x ) = \int _ { 2 } ^ { x } \left( 5 t ^ { 2 } - 2 t \right) d t ?$

Find $\int x ^ { 3 } - \frac { 5 } { x ^ { 6 } } + \frac { 2 } { x } - 4 e ^ { - 4 x } d x$

Which of the following represents the consumption function when $M P C = 0.4 + \frac { 0.1 } { \sqrt { Y } }$ , and $C = 50$ when $Y = 100$ ?

Find the consumer's surplus at Q = 8 for the demand function P = 50 - 4Q.

The producer's surplus at Q = a for the supply function P = 6 + 8Q Is known to be 400. Find the value of a.

Evaluate $\int _ { - 1 } ^ { 2 } 5 x ^ { 2 } - 4 x + 6 d x$

Which of the following expressions represents the present value of a continuous revenue stream for n years at a constant rate of $S per year if the discount rate is r%?

A firm's marginal revenue and marginal cost functions are given by MR = 240 - 0.6Q2 and MC = 150 + 0.3Q2 If fixed costs are 50, determine the maximum profit.

Quiz 6: Integration

Evaluate ∫2101(2x+5) (2x+5) dx\int _ { 2 } ^ { 10 } \frac { 1 } { ( 2 x + 5 ) \sqrt { ( 2 x + 5 ) } } d x∫210​(2x+5) (2x+5) ​1​dx

If the investment flow is I(t) =5000t14I ( t ) = 5000 t ^ { \frac { 1 } { 4 } }I(t) =5000t41​ calculate the capital formation from the end of the second year to half- way through the fifth year. Give your answer to the nearest whole number.

Which of the following is an expression for g'(x) when g(x) =∫2x(5t2−2t) dt?g ( x ) = \int _ { 2 } ^ { x } \left( 5 t ^ { 2 } - 2 t \right) d t ?g(x) =∫2x​(5t2−2t) dt?

Find ∫x3−5x6+2x−4e−4xdx\int x ^ { 3 } - \frac { 5 } { x ^ { 6 } } + \frac { 2 } { x } - 4 e ^ { - 4 x } d x∫x3−x65​+x2​−4e−4xdx

Which of the following represents the consumption function when MPC=0.4+0.1YM P C = 0.4 + \frac { 0.1 } { \sqrt { Y } }MPC=0.4+Y​0.1​ , and C=50C = 50C=50 when Y=100Y = 100Y=100 ?