Definite Integrals Questions and Answers

The marginal average cost of producing x digital sports watches is given by the function c x where C x is the average cost in dollars c x 1 400 2 Find the average cost function and the cost function What are the fixed costs C 100 25 The average cost function is C x The cost function is C x 0 The fixed costs are 1400

Calculate the definite integral by referring to the figure with the indicated areas Area C 5 856 Area D 1 71 Area A 1 549 Area B 2 734 to

41 42 Find equations of the normal plane and osculating plane of the curve at the given point BO smo ob impu 41 x 2 sin 3t y t z 2 cos 3t 0 2

1 4 Find the length of the curve 1 r t t 3 cos t 3 sin t 5 t 5 2 r t cos ti sin tj In cos tk 0 t T 4 3 r t i t j t k 0 t 1 3

y 4 6 45 45 4 6

4 2 1 t 1 dt

41 cos 3 To 2

Describe the features of the function f x 2log 1 The function has a 12 11 10 9 8 7 6 5 4 3 2 11 side is as 20005 and it is 11 6 6 6 i c b p E 2 3 4 5 6 7 8 9 10 11 12 asymptote of on its domain of The range of the function is The end behavior on the RIGH and the end behavior on the LEFT side is as

For the definite integral do the following 1 1 a Evaluate it using the table of integrals on the inside back cover Leave answers in exact form dx b Use a graphing calculator to verify your answer to part a Round your answer to three decimal plac 091

Speedometer readings for a motorcycle at 12 second intervals are given in the table t s 0 12 24 36 48 60 v ft s 32 27 24 22 24 27 a Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals b Give another estimate using the velocities at the end of the time periods c Are your estimates in parts a and b upper and lower estimates Explain b is a lower estimate and a is an upper estimate since v is a decreasing function of t a is a lower estimate and b is an upper estimate since v is an increasing function of t a and b are neither lower nor upper estimates since v is neither an increasing nor decreasing function of t

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function y y COS 2 S 10 1 25 0 dv

K L de 12 9 da 86

Consider the indefinite integral u ex 5 and e z e 5 This can be transformed into a basic integral by letting du e X 7 U dx Performing the substitution yields the integral n du S Integrating yields the result da

Use the Definition to find an expression for the area under the graph of f as a limit Do not evaluate the limit f x x 1 2x 6 x 8 lim 1248 3 Morrage instructor

If f 1 10 f is continuous and f 5 x d 17 what is the value of f 5

Find the des using the geometric distribution the Poisson distribution or the binomial distribution I hen determine if the events are unusual If convenient use the appropriate probability table or technology to find the The mean number of births per minute in a country in a recent year was about seven Find the probability that the number of births in any given minute is a exactly four b at least four and c more than four sled prc a P exactly four Round to four decimal places as needed b P at least four Round to four decimal places as needed c P more than four Round to four decimal places as needed Which of the events are unusual Select all that apply A The event in part a is unusual B The event in part b is unusual C The event in part c is unusual D None of the events are unusual

Use absolute value on a graphing calculator to find the area bounded by the graphs of the equations over the given interval y exy 0 11x 0 41 2 x 2 The area is Round to three cubic units units d

Find the consumers surplus and the producers surplus at the equilibrium price 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 70 0 04x p S x 20e0 001x What is the consumers surplus CS CS Round to the nearest integer as needed Round all intermediate steps to the nearest integer

Use a graphing calculator to find where the curves intersect and to find the area between the curves y ex y 3x 5x a The left point of intersection is Type integers or decimals rounded to the nearest thousandth as needed Type an ordered pair

Use the Fundamental Theorem of Calculus to find the derivative of I 1 f x f x dt 3 1 t5 1 1 x OF

20 points Let C be the curve obtained by going from 0 1 to 1 0 counter clockwise aroun e unit circle Let F x y x y e x y 1 2 e x y 1 2 Compute F F dr

An amusement company maintains records for each video game it installs in an arcade Suppose that C t and R t represent the total accumulated costs and revenues in thousands of dollars respectively t years after a particular game has been installed and the derivatives C t and R t are the following functions C t 3 and R t 8 0 3t Find the area between the graphs of C and R over the interval on the t axis from 0 to the useful life of the game and interpret the results What is the useful life of the game t years Round to the nearest tenth as needed

The future value at 7 interest compounded continuously for 5 years of the continuous income stream with rate c 0 02t flow f t 900e is 5 142 Compute the interest earned Type an integer or a decimal ECOD

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line y 2x x 2y about the y axis 107 2 V Sketch the region y 4 3 y f

The future value at 5 interest compounded continuously for 7 years of the continuous income stream with rate of 0 02t flow f t 2 100e is 16 491 Compute the interest earned Type an integer or a decimal

A company manufactures downhill skis It has fixed costs of 26 000 and a marginal cost given by C x 1 0 05x where C x is the total cost at an output of x pairs of skis Use a table of integrals to find the cost function C x and determine the production level to the nearest unit that produces a cost of 150 000 What is the cost to the nearest dollar for a production level of 850 pairs of skis Click the icon to view a brief table of integrals C x Simplify your answer Ilse inteners www fractions for any numbers in the expression

Use a table of integrals to find the consumers surplus at a price level of p 20 for the following price demand equation 14 000 40x p D x 400 x Click the icon to view a brief table of integrals The consumers surplus is 1 Round to the nearest dollar as needed

A car leasing firm must decide how much to charge for maintenance on the cars it leases After careful study the firm determines that the rate of maintenance M x on a new car will be approximately M x 42 1 x dollars per yea where x is the number of years the car has been in use What total maintenance cost can the company expect for a 2 year lease Whaf minimum amount should be added to the monthly lease payments to pay for maintenance on a 2 year lease Write a definite integral to find the total maintenance cost for a 2 year lease 10x dx The total maintenance cost the company can expect on a 2 year lease is Round to the nearest cent as needed The minimum amount that should be added to the monthly lease payments is Round to the nearest cent as needed

r 8 sin 0 is equivalent to what rectangular equation OA x y 4 OC x 8 y 8 www OB x 4 y 4 OD x y 4 4

12 S 9 1 2

599e Bay 3 71 dy Y

Find the area bounded by the graphs of the indicated equations over the given interval when stated Compute answers to three decimal places y x 3 y 6x 6 1 x 2 The area calculated to three decimal places is ACTOR square units

Find the area bounded by the graphs of the indicated equations over the given interval when stated Compute answers to three decimal places y x 3 y 6x 6 1 x 2 COFFR The area calculated to three decimal places is square units

A yeast culture is growing at a rate of W t 0 3e0 09 grams per hour Find the area between the graph of W and the t axis over the interval 0 10 and interpret your results The area under the curve rounded to three decimal places is weight gain over ten hours will be This means that the CEES square units grams of yeast 4 865

Monthly sales of a particular personal computer are expected to decline at a rate of S t 4te0 2t computers per month where tis time in months and S t is the number of computers sold each month The company plans to stop manufacturing this computer when monthly sales reach 800 computers If monthly sales now t 0 are 2 000 computers find S t How long to the nearest month will the company continue to manufacture the computer S t CECH The company will continue to manufacture the computer for approximately months

Choose the correct answer below OA P x k x dx OC Sk x dx C d OE SIP x K X k x dx OB S p x k x dx d OD k x p x dx OF Sp x dx y k x y p x cd

Evaluate 5 1 2x 9 x 9x 5 dx 5 S 1 Type an exact answer 2x 9 x 9x 5 dx

Use the trapezoidal rule with n 3 to approximate 9 x4 dx T3 Round the final answer to two decimal places as needed Round all intermediate values to four decimal pl as needed

The concentration of particulate matter in parts per million t hours after a factory ceases operation for the day is given by the following formula C t Find the average concentration for the period from t 0 to t 4 C t 8In t 9 t 9 The average concentration of particulate matter for the time period from t 0 to t 4 is approximately million Do not round until the final answer Then round to four decimal places as needed parts per

Use absolute value on a graphing calculator to find the area between the curve and the x axis over the given interval 13 y x ex 2 x 2 The area between the curve and the x axis over the given interval is Type an integer or decimal rounded to two decimal places as needed

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

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

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

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

Does 4n6 n 1 n7 1 converge or diverge

ind the exact length of the curve x 5 6t y 4 4t 0 t 4

2 20 points Consider the cone V whose base is the unit disk in the xy plane and whose height is 1 along z axis as shown below The dimensions are measured in meters 1 0 1 0 0 5 8 50 Z 0 5 1 0 0 0 0 5 1 0 The cone is made out of a material of variable density at a point x y z the density will be 8 x y z 100 x y kilograms per cubic meter Compute the total mass of this cone in kilograms Hint the total mass will be the integral of the density over all of V

Find the area enclosed by the graphs of sin x and cos x between x 0 and shown below 1 Question Help Read 2 3

Find the arclength of y 3x 2 on 2 x 3 16 15 14 13 12 H 10 9 78 Give your answer to at least two decimal places of accuracy

Find the area between y x 1 and y x 2 Round your limits of integration and answer to 2 decimal places 10 9 7 6 5 q The area between the curves is