Calculus Questions

Consider the function f t 4 sec t 2t Let F t be the antiderivative of f t with F 0 0 Then F 5 4 tan 2 2 6

cal integration command on a graphing calculator Find the future value at 9 75 interest compounded continuously for 10 years of the continuous income stream with the rate of function f t 700t 11 000 GLOOB The future value of the income stream at 9 75 compounded continuously at the end of 10 years is Round to the nearest dollar as needed 0

Consider the function f t 6 sec t 7t Let F t be the antiderivative of f t with F 0 0 Then F 2 6 tan 2 21 X

8 3 Does E 1 2 n 0 37 converge or diverge If it converges what is the sum

Find the derivative of 2e 92 cos 6x Hint use product rule and chain rule Use e x for e 12e 9x Y 9x sin 6x 18e cos 6x Now find the equation of the tangent line to the curve at X 0 Write your answer in ma b format

Q what are the bounds of integration for the first integral and a f x da f x dx f a da f x dx F x dx f x dx 5

Consider the function f t 6 sec t 9t Let F t be the antiderivative of f t with F 0 0 Then F 2

5 If x 2t t and y 6 2t find an equation of the tangent in terms of x and y to e curve at t 2

Write the first five terms of the geometric sequence an 4an a 2 OA 8 32 128 512 2 048 OB 4 8 32 128 512 O C OD 2 2 6 10 14 2 8 32 128 512

Find the sum of the first 9 terms of the following geometric sequence 8 16 32 64 128 OA 4 090 OB 4 068 OC 4 125 OD 4 088

4 Determine the domain Express your answer in interval notation 5 points f x log 2 3

Write the first five terms of the arithmetic sequence a 2 d 2 A 2 4 6 8 10 4 6 8 10 12 0 2 4 6 8 OD 2 3 4 5 6 OB O C

3 The half life of a radioactive substance is 20 years If we start with 15 gram a how much is left after 5 years b how long will it take to have 5 grams left

5 Solve the equation Do not use a calculator Give an exact value 5 points 1 4 ln 2z 1 5

1 Write an equivalent statement in logarithmic form 4 points 1 100 10 000

The tables to the right give price demand and price supply data for the sale of soybeans at a grain market where x is the number of bushels of soybeans in thousands of bushels and p is the price per bushel in dollars Use quadratic regression to model the price demand data and linear regression to model the price supply data Complete parts A and B below A Find the equilibrium quantity and equilibrium price The equilibrium quantity is thousand bushels Round to three decimal places as needed Price Demand p D x 6 64 6 54 6 45 6 38 6 35 X 0 10 20 30 40 Price Supply P S x 6 34 6 40 6 45 6 48 6 53 X 0 10 20 30 40

3 Solve the equation Give an exact value AND an approximation to the nearest thousandth 5 points 2 3 5

b lim X XX x 1 X

2 Rewrite as one single logarithm with coefficient 1 4 points log 2y 5 2 log y 3

Find the consumers surplus and the producers surplus at the equlibrium level for the given price demand and price supply equations Include a graph that identifies the consumers surplus and the producers surplus Round all values to the nearest integer p D x 36 0 06x p S x 11 0 04x The value of x at equilibrium is 250 The value of p at equilibrium is 21 The consumers surplus at equilibrium is

In order to study the effects of World War II on a certain country s economy an economist used data from that country s census bureau to produce the following Lorenz curves for the distribution of income in 1935 and 1947 f x x25 Lorenz curve for 1935 16 g x x 6 Lorenz curve for 1947 Find the Gini index of income concentration for each Lorenz curve and interpret the results What is the Gini index for 1935 0 428 Round to three decimal places as needed What is the Gini index for 1947 Round to three decimal places as needed

Find the producers surplus at a price level of p 58 for the price supply equation below p S x 5 0 1x 0 0003x The producers surplus is Round to the nearest intogor pooded

F 2x 12x x 4x 12 3 4 8x 24x x 9x 18

A C x2 5x 6 x 3x x 2x x 2x 1 X 3 x 4x 3 x 3x B D y2 7y 1 y 8y 7 x 2x 3x 2x x 2 x 1

Find the derivative of the function g x g x et 2 et 5 2x

2 6x 3 Find the constant term in the expansion of 6x

In 2012 34 1 of all glass containers were recycled A beverage company used 350 000 lb of glass containers per year After recycling the amount of glass in pounds still in use after t years is given by N t 350 000 0 373 a Find N 7 and explain its meaning b Find N 7 and explain its meaning c When will 6 of the original amount of glass still be in use a N 7 351 58 lb Round to two decimal places as needed The expression N 7 means that after year s the amount of glass still in use N 7 pounds N 7 pounds per year N 7 pounds per year N 7 pounds

An office machine is purchased for 7300 Assume that its salvage value V in dollars depreciates according to a method called double declining balance by 30 each year and is given by V t 7300 0 70 where t is the time in years after purchase a Find V 3 and explain its meaning b Find V 3 and explain its meaning c When will the salvage value of the office machine be half of the purchase price a V 3 2503 9 Round to the nearest cent as needed The expression V 3 means that after 3 year s the salvage value b V 3 Round to the nearest cent as needed is V 3 dollars

A scientist has two cultures of bacteria Culture A starts with a volume of 3 ml and doubles every 4 hours Culture B starts with a volume of 2 ml and doubles every 3 hours What should be the starting volume of culture A if we want the two cultures to have the same volume after 12 hours O a 2 O b 3 O c 4 O d 5 O e 6

In some chemical reactions the rate at which the amount of a substance changes with time is proportional to the amount present For example this is the case as 8 glucono lactone changes into gluconic acid a Which of the following differential equations is satisfied by y the quantity of 8 glucono lactone present at time for some constant k dy dt O dy dt dy dt O dy dt k y ky 11 ky eTextbook and Media 4 hours

The equation A t 8000e0 055t gives the balance after t years of an initial investment of 8000 dollars which pays 5 50 compounded continuously dA a Find a formula for dt b Find and interpret A 10 Include appropriate units c Compare the approximation of 763 to the actual change Report your answer to two decimal places a dA dt 440e0 055t b A 10 Round up to the r dollars increased in the total amount per yearly increase in the investment time period years per dollar increase in the initial investment dollars years increased in the total amount per dollar increase in the initial investment

Find the area of the region bounded by the graphs of the given equations y 4x y x x 0 x 2 O A B S Ox f dx SC 3 dx dx 0 5

A company s total cost in millions of dollars is given by C t 100 40et where t is the time in years since the start up date The graph of C t is shown to the right Find each of the following a The marginal cost C t b C 0 c C 5 d Find lim C t and lim C t t 0 t a C t 40et Do not include the symbol in your answer b C 0 40 million per year Simplify your answer Do not include the symbol in your answer Ay 150 75 15 30 c C 5 per year Simplify your answer Round to the nearest thousand as needed Do not include the symbol in your answer 57

7 Find the Cartesian coordinates of the polar point 2 3

Find s t where s t represents the position function v t represents the velocity function and a t represents the acceleration function a t 12t 4 with v 0 8 and s 0 1 s t

In 2012 34 1 of all glass containers were recycled A beverage company used 350 000 lb of glass containers per year After recycling the amount of glass in pounds still in use after t years is given by N t 350 000 0 373 a Find N 7 and explain its meaning b Find N 7 and explain its meaning c When will 6 of the original amount of glass still be in use a N 7 lb Round to two decimal places as needed

Find the area bounded by y 2x3 54x 135 the x axis and the first coordinates of the relative maximum and minimum values of the function The area is

2 Evaluate fx cos x dx as an infinite series

Use the fact that the derivative gives the slope of a curve to decide which of the graphs A F in the figure below could represent a solution to the differential equation A C E 6x B xe LV D H F XA y I

dy Is there a value of n which makes y x a solution to the equation 3x y If so what is the exact value dx Enter NA if there is no value for n n

Find the Taylor series for f x sin x centered at a EIN

Find the Taylor series for f x sin a centered at a 11 kla

Determine which of the following functions is a solution to the differential equation Oy 4t Oy 3t Oy 14 Oy 4 Oy 1 dy dt 4y

1 Find the Taylor series for f x sin x centered at a KIN

A motorcyclist being monitored by radar accelerates at a constant rate from 0 mph v 0 0 to 55 mph in 15 sec How far has the motorcycle traveled after 15 sec Hint Convert seconds to hours After 15 sec the motorcycle has traveled mi Simplify your answer Type an integer or a fraction

D x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and producer surplus at the equilibrium point D x x 9 S x x 4x 15 a What are the coordinates of the equilibrium point 3 36 Type an ordered pair b What is the consumer surplus at the equilibrium point 63 Round to the nearest cent as needed c What is the producer surplus at the equilibrium point Round to the nearest cent as needed

Find the derivative of the function g x g x e 3x 1 4 3x Pe 4 3x

Does n 1 n 5 converge or n 10n 9 2 diverge 16

Does 4n6 n 1 n7 1 converge or diverge

O x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point D x x 9 S x x 4x 15 a What are the coordinates of the equilibrium point 3 36 Type an ordered pair b What is the consumer surplus at the equilibrium point Round to the nearest cent as needed