Question 1

Find the general solution of $\frac { d y } { d x } = e ^ { x }$ .

Accepted Answer

A)    y=xeCy = \frac { x ^ { e } } { C }y=Cxe​ 
B)    y=xex+Cy = x e ^ { x } + Cy=xex+C 
C)    y=Cexy = C e ^ { x }y=Cex 
D)    y=ex+Cy = e ^ { x } + Cy=ex+C D

Question 2

Compute $f _ { x }$ for $f ( x , y ) = 7 x y ^ { 2 }$ .

Accepted Answer

A)    14xy114 x y ^ { 1 }14xy1 
B)  7x
C)    14xy1+7y214 x y ^ { 1 } + 7 y ^ { 2 }14xy1+7y2 
D)    7y27 y ^ { 2 }7y2 D

Question 3

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$

Accepted Answer

A)  Diverges.
B)  Converges to  52\frac { 5 } { 2 }25​ 
C)  Converges to  512\frac { 5 } { 12 }125​ 
D)  Converges to  72\frac { 7 } { 2 }27​ B

Question 4

To raise money,a service club has been collecting used bottles that it plans to deliver to a local glass company for recycling.Since the project began 90 days ago,the club has collected 45,000 pounds of glass for which the glass company currently offers 1 cent per pound.However,because bottles are accumulating faster than they can be recycled,the company plans to reduce by 1 cent each day the price it will pay for 100 pounds of used glass.Assume that the club can continue to collect bottles at the same rate and that transportation costs make more than one trip to the glass company unfeasible.What is the most advantageous time for the club to conclude its project and deliver the bottles?

Accepted Answer

A)  15 days from now
B)  Today
C)  5 days from now
D)  55 days from now

Question 5

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = 28 x ^ { 3 } y ^ { 2 }$ ; y = 6 when x = 1.

Accepted Answer

A)    y=643−42x4y = \frac { 6 } { 43 - 42 x ^ { 4 } }y=43−42x46​ 
B)    y=643+42x4y = \frac { 6 } { 43 + 42 x ^ { 4 } }y=43+42x46​ 
C)    y=−643+42x4y = - \frac { 6 } { 43 + 42 x ^ { 4 } }y=−43+42x46​ 
D)    y=643−42x3y = \frac { 6 } { 43 - 42 x ^ { 3 } }y=43−42x36​

Question 6

Find the equation of the tangent line to f (x)= ln x + 2 at x = 1.

Accepted Answer

A)  y = 1
B)  x = 1
C)  y = x + 1
D)  y = x

Question 7

Use a double integral to find the area of R. R is the region bounded by y = 3x,y = ln x,y = 0,and y = 1.

Accepted Answer

A)    ∫y−0y−1∫x−y3x−ey1dxdy=e−76\int _ { y - 0 } ^ { y - 1 } \int _ { x - \frac { y } { 3 } } ^ { x - e ^ { y } } 1 d x d y = e - \frac { 7 } { 6 }∫y−0y−1​∫x−3y​x−ey​1dxdy=e−67​ 
B)    ∫y−0y−1∫x−0x−ey1dxdy=e−1\int _ { y - 0 } ^ { y - 1 } \int _ { x - 0 } ^ { x - e ^ { y } } 1 d x d y = e - 1∫y−0y−1​∫x−0x−ey​1dxdy=e−1 
C)    ∫x=0x−e∫y−ln⁡xy−11dydx=e\int _ { x = 0 } ^ { x - e } \int _ { y - \ln x } ^ { y - 1 } 1 d y d x = e∫x=0x−e​∫y−lnxy−1​1dydx=e 
D)    ∫x=0x−2∫y−m⁡xy−3x1dydx=3e22\int _ { x = 0 } ^ { x - 2 } \int _ { y - \operatorname { m } x } ^ { y - 3 x } 1 d y d x = \frac { 3 e ^ { 2 } } { 2 }∫x=0x−2​∫y−mxy−3x​1dydx=23e2​

Question 8

Suppose that during business hours,the time between successive wireless calls at a network switch can be represented by an exponentially distributed random variable X with expected value $E ( X ) = 0.025$ second.Using an appropriate probability density function,find the probability that the time between the arrival of successive wireless calls at the switch is between 0.03 and 0.065 second.Round to the nearest hundredth.

Accepted Answer

A)  0.63
B)  0.46
C)  0.77
D)  0.23

Question 9

A study shows that consumers in one city will buy q hundred bottles of a new perfume when the price is $p = 30 - 8 q + \sin \left( \frac { \pi q } { 3 } \right)$ dollars per bottle.Find the consumer's surplus for this product when 22,000 bottles are demanded and produced.Round to the nearest dollar.

Accepted Answer

A)  $2,311
B)  $1,932
C)  $1,837
D)  $847

Question 10

Write a differential equation describing the given situation: Powdered lemonade dissolves in a pitcher at a rate that is proportional to the amount of undissolved powder remaining.Let P = the total amount of powder added,D = the amount dissolved,and t = time.

Accepted Answer

A)    dDdt=k+P−D\frac { d D } { d t } = k + P - DdtdD​=k+P−D 
B)    dDdt=k(P−
D)  \frac { d D } { d t } = k ( P - D ) dtdD​=k(P−
D)   
C)    dDdt=kP(P−
D)  \frac { d D } { d t } = k P ( P - D ) dtdD​=kP(P−
D)   
D)    dDdt=kPD\frac { d D } { d t } = k P DdtdD​=kPD

Question 11

Find the general solution of the given first-order linear differential equation. $x ^ { 3 } \frac { d y } { d x } + x ^ { 2 } y = 3$

Accepted Answer

A)    y=Cx2−3xy = \frac { C } { x ^ { 2 } } - \frac { 3 } { x }y=x2C​−x3​ 
B)    y=Cx−3x2y = \frac { C } { x } - \frac { 3 } { x ^ { 2 } }y=xC​−x23​ 
C)    y=C−3x2y = C - \frac { 3 } { x ^ { 2 } }y=C−x23​ 
D)    y=−Cx2y = - \frac { C } { x ^ { 2 } }y=−x2C​

Question 12

Find the relative rate of change of f (x)with respect to x for the prescribed value x = 1. f (x)= 3x3 + x2 - 8

Accepted Answer

A)    111\frac { 1 } { 11 }111​ 
B)    −114- \frac { 11 } { 4 }−411​ 
C)  11
D)    −411- \frac { 4 } { 11 }−114​

Question 13

Suppose that the number of people suffering bee stings in a rural county in May follows a Poisson distribution with a mean of 1.5 bee stings per day.Find the probability that on a randomly selected day,there will be 2 bee stings.Round to three decimal places.

Accepted Answer

A)  0.151
B)  0.251
C)  0.201
D)  0.100

Question 14

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = \ln ( 5 x ) ; a = \frac { 1 } { 5 }$

Accepted Answer

A)    ∑n=1∞(−1) n+15nn(x−15) n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 )  ^ { n + 1 } 5 ^ { n } } { n } \left( x - \frac { 1 } { 5 } \right)  ^ { n }∑n=1∞​n(−1) n+15n​(x−51​) n 
B)    ∑n=1∞5nn(x−15) n\sum _ { n = 1 } ^ { \infty } \frac { 5 ^ { n } } { n } \left( x - \frac { 1 } { 5 } \right)  ^ { n }∑n=1∞​n5n​(x−51​) n 
C)    ∑n=1∞1n5n(x−15) n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n 5 ^ { n } } \left( x - \frac { 1 } { 5 } \right)  ^ { n }∑n=1∞​n5n1​(x−51​) n 
D)    ∑n=1∞(−1) n+1n5n(x−15) n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 )  ^ { n + 1 } } { n 5 ^ { n } } \left( x - \frac { 1 } { 5 } \right)  ^ { n }∑n=1∞​n5n(−1) n+1​(x−51​) n

Question 15

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t). $$f ( t ) = 1.5 + 5.5 \sin \left[ \frac { \pi } { 3 } ( t - 1 ) \right]$$

Accepted Answer

A)    p=6,b=−5.5,d=−1,a=1.5p = 6 , b = - 5.5 , d = - 1 , a = 1.5p=6,b=−5.5,d=−1,a=1.5 
B)    p=3,b=1.5,d=1,a=5.5p = 3 , b = 1.5 , d = 1 , a = 5.5p=3,b=1.5,d=1,a=5.5 
C)    p=6,b=5.5,d=1,a=1.5p = 6 , b = 5.5 , d = 1 , a = 1.5p=6,b=5.5,d=1,a=1.5 
D)    p=6,b=1.5,d=1,a=5.5p = 6 , b = 1.5 , d = 1 , a = 5.5p=6,b=1.5,d=1,a=5.5

Question 16

Evaluate $\int e ^ { 6 x - 8 } d x$ .

Accepted Answer

A)    (6x−8) e6x−8+C( 6 x - 8 )  e ^ { 6 x - 8 } + C(6x−8) e6x−8+C 
B)    (6x−8) Ce6x−8( 6 x - 8 )  C e ^ { 6 x - 8 }(6x−8) Ce6x−8 
C)    e6x−8+Ce ^ { 6 x - 8 } + Ce6x−8+C 
D)    e6x−86+C\frac { e ^ { 6 x - 8 } } { 6 } + C6e6x−8​+C

Question 17

Records indicate that t hours past midnight,the temperature at the local airport was $f ( t ) = - 0.3 t ^ { 2 } + 4 t + 10$ degrees Fahrenheit.What was the average temperature at the airport between 8:00 A.M.and noon? Round your answer to one decimal place,if necessary.

Accepted Answer

A)  3.9 degrees F
B)  9.6 degrees F
C)  98.0 degrees F
D)  19.6 degrees F

Question 18

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { 7 ^ { k - 1 } x ^ { k } } { k }$

Accepted Answer

A)    R=R =R=  7
B)    R=R =R=   ∞\infty∞ 
C)    R=R =R=   17\frac { 1 } { 7 }71​ 
D)    R=R =R=  0

Question 19

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .

Accepted Answer

A)    −3e−2x+C- 3 e ^ { - 2 x } + C−3e−2x+C 
B)    32xe−2x+C\frac { 3 } { 2 } x e ^ { - 2 x } + C23​xe−2x+C 
C)    e−2x(32x−14) +Ce ^ { - 2 x } \left( \frac { 3 } { 2 } x - \frac { 1 } { 4 } \right)  + Ce−2x(23​x−41​) +C 
D)    xe−2x+Cx e ^ { - 2 x } + Cxe−2x+C

Question 20

Differentiate the given function. $y = 5 \tan ( 2 x - \sqrt { x } )$

Accepted Answer

A)    (10−52x) sec⁡2(2x−x) \left( 10 - \frac { 5 } { 2 \sqrt { x } } ight)  \sec ^ { 2 } ( 2 x - \sqrt { x } ) (10−2xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​5​) sec2(2x−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​)  
B)    5sec⁡2(2x−x) 5 \sec ^ { 2 } ( 2 x - \sqrt { x } ) 5sec2(2x−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​)  
C)    5cot⁡(12x−x) 5 \cot \left( \frac { 1 } { 2 x - \sqrt { x } } ight) 5cot(2x−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​1​)  
D)    (10−52x) tan⁡(2x−x) sec⁡(2x−x) \left( 10 - \frac { 5 } { 2 \sqrt { x } } ight)  	an ( 2 x - \sqrt { x } )  \sec ( 2 x - \sqrt { x } ) (10−2xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​5​) tan(2x−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​) sec(2x−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"/>​)

Exam 13: Mathematical Problem Set: Calculus, Algebra, and Optimization

Find the general solution of $\frac { d y } { d x } = e ^ { x }$ .

Compute $f _ { x }$ for $f ( x , y ) = 7 x y ^ { 2 }$ .

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = 28 x ^ { 3 } y ^ { 2 }$ ; y = 6 when x = 1.

Find the equation of the tangent line to f (x) = ln x + 2 at x = 1.

Use a double integral to find the area of R. R is the region bounded by y = 3x,y = ln x,y = 0,and y = 1.

A study shows that consumers in one city will buy q hundred bottles of a new perfume when the price is $p = 30 - 8 q + \sin \left( \frac { \pi q } { 3 } \right)$ dollars per bottle.Find the consumer's surplus for this product when 22,000 bottles are demanded and produced.Round to the nearest dollar.

Write a differential equation describing the given situation: Powdered lemonade dissolves in a pitcher at a rate that is proportional to the amount of undissolved powder remaining.Let P = the total amount of powder added,D = the amount dissolved,and t = time.

Find the general solution of the given first-order linear differential equation. $x ^ { 3 } \frac { d y } { d x } + x ^ { 2 } y = 3$

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) = 3x³ + x² - 8

Suppose that the number of people suffering bee stings in a rural county in May follows a Poisson distribution with a mean of 1.5 bee stings per day.Find the probability that on a randomly selected day,there will be 2 bee stings.Round to three decimal places.

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = \ln ( 5 x ) ; a = \frac { 1 } { 5 }$

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t) . $f ( t ) = 1.5 + 5.5 \sin \left[ \frac { \pi } { 3 } ( t - 1 ) \right]$

Evaluate $\int e ^ { 6 x - 8 } d x$ .

Records indicate that t hours past midnight,the temperature at the local airport was $f ( t ) = - 0.3 t ^ { 2 } + 4 t + 10$ degrees Fahrenheit.What was the average temperature at the airport between 8:00 A.M.and noon? Round your answer to one decimal place,if necessary.

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { 7 ^ { k - 1 } x ^ { k } } { k }$

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .

Differentiate the given function. $y = 5 \tan ( 2 x - \sqrt { x } )$

Exam 13: Mathematical Problem Set: Calculus, Algebra, and Optimization

Find the general solution of dydx=ex\frac { d y } { d x } = e ^ { x }dxdy​=ex .

Compute fxf _ { x }fx​ for f(x,y) =7xy2f ( x , y ) = 7 x y ^ { 2 }f(x,y) =7xy2 .

Determine whether the given geometric series converges,and if so,find its sum. ∑n=1∞(57) n\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }∑n=1∞​(75​) n

Find the particular solution of the given differential equation that satisfies the indicated condition: dydx=28x3y2\frac { d y } { d x } = 28 x ^ { 3 } y ^ { 2 }dxdy​=28x3y2 ; y = 6 when x = 1.

Find the equation of the tangent line to f (x) = ln x + 2 at x = 1.

Use a double integral to find the area of R. R is the region bounded by y = 3x,y = ln x,y = 0,and y = 1.

Write a differential equation describing the given situation: Powdered lemonade dissolves in a pitcher at a rate that is proportional to the amount of undissolved powder remaining.Let P = the total amount of powder added,D = the amount dissolved,and t = time.

Find the general solution of the given first-order linear differential equation. x3dydx+x2y=3x ^ { 3 } \frac { d y } { d x } + x ^ { 2 } y = 3x3dxdy​+x2y=3

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) = 3x3 + x2 - 8

Suppose that the number of people suffering bee stings in a rural county in May follows a Poisson distribution with a mean of 1.5 bee stings per day.Find the probability that on a randomly selected day,there will be 2 bee stings.Round to three decimal places.

Find the Taylor series for the given function at the indicated point x=a. x = a \text {. }x=a. f(x) =ln⁡(5x) ;a=15f ( x ) = \ln ( 5 x ) ; a = \frac { 1 } { 5 }f(x) =ln(5x) ;a=51​

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t) . f(t) =1.5+5.5sin⁡[π3(t−1) ]f ( t ) = 1.5 + 5.5 \sin \left[ \frac { \pi } { 3 } ( t - 1 ) \right]f(t) =1.5+5.5sin[3π​(t−1) ]

Evaluate ∫e6x−8dx\int e ^ { 6 x - 8 } d x∫e6x−8dx .

Determine the radius of convergence for the given power series. ∑k=0∞7k−1xkk\sum _ { k = 0 } ^ { \infty } \frac { 7 ^ { k - 1 } x ^ { k } } { k }∑k=0∞​k7k−1xk​

Evaluate ∫(2−3x) e−2xdx\int ( 2 - 3 x ) e ^ { - 2 x } d x∫(2−3x) e−2xdx .

Differentiate the given function. y=5tan⁡(2x−x) y = 5 \tan ( 2 x - \sqrt { x } ) y=5tan(2x−x​)

Find the general solution of $\frac { d y } { d x } = e ^ { x }$ .

Compute $f _ { x }$ for $f ( x , y ) = 7 x y ^ { 2 }$ .

Determine whether the given geometric series converges,and if so,find its sum. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 } { 7 } \right) ^ { n }$

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = 28 x ^ { 3 } y ^ { 2 }$ ; y = 6 when x = 1.

Find the general solution of the given first-order linear differential equation. $x ^ { 3 } \frac { d y } { d x } + x ^ { 2 } y = 3$

Find the relative rate of change of f (x) with respect to x for the prescribed value x = 1. f (x) = 3x³ + x² - 8

Find the Taylor series for the given function at the indicated point $x = a \text {. }$ $f ( x ) = \ln ( 5 x ) ; a = \frac { 1 } { 5 }$

Determine the period p,the amplitude b,the phase shift d,and the vertical shift a of the given trigonometric function f(t) . $f ( t ) = 1.5 + 5.5 \sin \left[ \frac { \pi } { 3 } ( t - 1 ) \right]$

Evaluate $\int e ^ { 6 x - 8 } d x$ .

Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { 7 ^ { k - 1 } x ^ { k } } { k }$

Evaluate $\int ( 2 - 3 x ) e ^ { - 2 x } d x$ .

Differentiate the given function. $y = 5 \tan ( 2 x - \sqrt { x } )$