Calculus Questions

among circular functions to find cos 1 22 7 50 2 1 0 2x 3x 3 49 2 4 4 NO MIN 0 AY 0 1 3x Ne KI3 wa 0 1 MI4 5x 4 7x 8 KIO 11x 3 F 1 0 X Q COS 3 Type an integer a exact answer usin

cot 6 Select the correct choice below and if necessary fill in the answer box to complete your choice O A K 6 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression B The expression is undefined cot 42 1 0 7x 4 1 42 42 45 3x 5x 4 6 65x 2x 4x 3 K2 0 1 3x 2 0 x y 1

Determine the values of the trigonometric functions of a tan Type a simplified fraction 25

the trigonometric function at the indicated real number s or state that the expression is undefined 11x 6 sec 42 42 1 0 7x 4 2 2 2 0 1 2x 3 3x 42 2 5x 4 6 6 5x 4x 3 K 2 73 5x 3 74 4 x 6 0 11x 1 0 3x 2 0 1 9 2 4 x y 1 7x 6 4 60 fill in the answer box to comple O A 11x 6 Simplify your answer radicals Use integers C numbers in the express B The expression is under sec

Use a in radian mode to find approximations for cos s and sins for the number s exact values they are only approximations Then determine the quadrant in which the point or corresponding to s lies Finally find approximations for tan s cots secs and csc s sin 1 27 Round to three decimal places

sin 5 4121 0 7650 Round to four decimal places cos 5 4121 Round to four decimal places

Use transformations of the graph of f x to identify the graph of the function given below Use the graph to determine its domain range and asymptote g x 1 Identify the graph of g x e 1 OA What is the domain of g x 12 Use interval notation What is the range of g x e 1 Use interval notation What line is the horizontal asymptote of g x 1 OB SITE

S 2 Select the correct choice below and if necessary fill in the answer box to complete your c 7x 2 B The function is undefined OA sin Type an integer or a fraction

K I Determine what fraction of the circumference of the unit circle the value of s represents For exa 1 represents of the circumference of the unit circle Do not use a calculator 2 I represents of the circumference of the unit circle 4 Type an integer or a simplified fraction

1 represents of the circumference of the unit circle

Suppose the average yearly salary of an individual whose final degree is a master s is 55 thousand less than twice that of an individual whose final degree is a bachelor s Combined two people with each of these educational attainments earn 116 thousand Find the aw yearly salary of an individual with each of these final degrees GED The average yearly salary for an individual whose final degree is a bachelor s is thousand and the average yearly salary for an individual whose final degree is a master s is thousand

The conect expression is 3 Type an equation using as the

ert the following degree measure to radians 90 radians Simplify your answer Type an exact answer using z

51 The complement of the angle is 39 Simplify your answer The supplement of the angle is Simplify your answer

The angle 120 corresponds to 0 of a complete rotation Type an integer or a simplified fraction

39x 1 x 9 Simplify your answers Type an integer or a fraction C 39x

The angle is Simplify your answer Type an integer or a simplified fraction 9 B

angle of 360 has an equivalent radian measure of C

oblem 6 Evaluate the following sum n 1 2i A n 4n B n 4n 10 C n 5m 15n 5 D n 4n 1 E n 5n 13 F n 5n 18n 2 G n 5n 15n 1 H n 4n 4

2x 16x Simplify your answers Type an integer or a fraction

Problem 5 a Suppose that f x x 3 for x 0 and that f 1 1 Find f 4

Determine the surface area of the solid obtained by rotating the following parametric curve about the x axis x cos 0 y sin 0 0 0 2

roblem 1 A box with an open top is to be constructed from a square piece of cardboard 4 ft wide by cutting out a square from each of the four corners and bending up the sides Find the largest volume that such a box can have

Determine the surface area of the solid obtained by rotating the following parametric curve about the x axis x cos 0 y sin 0 0 0

A company s marginal cost function is MC x xe x 2 and fixed costs are 215 Find the cost function Hint Evaluate the consta C so that the cost is 215 at x 0 C x 2e 5 4e 219 X

Question 3 60 points 3 Integrate each of the given expressions a 21 3 2 5 x 5 x 7 dx b 2 1 21 1 1 dt t2 c 2 x da d z 3 4z dx

Question 1 20 points 1 Determine the value of a that makes F x an antiderivative function of the given function f x a f x 5x F x ax5 b f x 10x 4 F x ax5 4 Paragraph V BI U A A v

a Replace the polar equation 8rsine with the equivalent Cartesian equation t

Question 2 20 points 2 Find antiderivatives F x of the given functions a f x 2x x b f x x x x 3 Paragraph V B I U A E E O 40 GE

Find the length of the parametrically defined curve x 8 cost 8t sint y 8 sin t 8t cost 0 t Use the equation editor to type your answer L

Find the area of the region inside the lemniscate r 6 cos 20 and outside the circle r 3 Points will be awarded based on uploaded work 71 6 21 3 411 3 2 11 2 3m 2 11 3 5m 3 71 6 11 6 0 1

Find the length of the spiral curve 0 0 T

a Replace the polar equation 8rsine with the equivalent Cartesian equation t

a Find the tangent to the curve x t and y t 4 at the point 2 3 where t 4 b Find as a function of t for the given x and y d y dx

In Exercises 13 22 find the limit of each rational function a as x and b as x 13 f x 2x 3 x 12 547 x 1 15 f x 2 17 h x 19 g x 21 h x x3 3 7x 3x 6x 10x5 x 31 6 2r 3r 2x 3 5x 3x

0 0 0 30 1354 9 11 13 15

You are a lifeguard and spot a distressed child 70 meters along the shore and 40 meters from the shore to the child You run along the shore and for a while and then jump into the water and swim from there directly to child You can run at a rate of 4 meters per second and swim at a rate of 1 2 meters per second How far along the shore should you run before jumping into the water in order to get to the child in the least amount of time Round your answer to three decimal places Hint Consider this diagram where a represents the distance running and y represents the distance swimming Try to write y in terms of x and use that to build the time as a function of x child you X 70 y 40

A fence 9 feet tall runs parallel to a tall building at a distance of 3 ft from the building as shown in the diagram e LADDER 9 ft 3 ft We wish to find the length of the shortest ladder that will reach from the ground over the fence to the wall of the building A First find a formula for the length of the ladder in terms of 0 Hint split the ladder into 2 parts Type theta for 0 L 0 B Now find the derivative L 0 Type theta for 0 L 0 C Once you find the value of that makes L 0 0 substitute that into your original function to find the length of the shortest ladder Give your answer

Find the absolute maximum and minimum values of f x x 6 x 5 Give your answers as a single value e g 17 or a comma separated list e g 1 2 3 or NONE The absolute maximum value is The absolute minimum value is f 1 if any over the interval 5 0 and it occurs at z and it occurs at z

10 5 10 LO Af x 15 X 10 C

Find and simplify the expression if f x x 3 f x 2 f x 2 Cimplic

Suppose that the monthly cost in dollars of producing x chairs is C x 0 007x3 0 07x 2 19x 700 and currently 40 chairs are produced monthly a What is the current monthly cost b What is the marginal cost when x 40 c Use the result from part b to estimate the monthly cost of increasing production to 42 chairs per month d What would be the actual additional monthly cost of increasing production to 42 chairs monthly a The current monthly cost is Round to the nearest cent as needed b The marginal cost when x 40 is per item Round to the nearest cent as needed c Use the result from part b to estimate the monthly cost of increasing production to 42 chairs per month The monthly cost in increasing production to 42 chairs monthly is Round to the nearest cent as needed d The actual additional monthly cost of increasing production to 42 chairs monthly is Round to the nearest cent as needed

Social animals that live in groups can spend less time scanning for predators than solitary individuals However they waste time fighting with the other group members over the available food There is some group size at which the net benefit is greatest because the animals spend least time on these unproductive activities and thus can spend time on feeding mating etc Assume that for a group of size x the fraction of time spent scanning for predators is 1 x 1 and the fraction of time spent fighting with other animals over food is F x B x 1 S x A where A B are constants and A 2B Find the size of the group for which the time wasted on scanning plus time wasted on fighting is smallest

For the demand function q D x 700 X find the following a The elasticity b The elasticity at x 1 stating whether the demand is elastic inelastic or has unit elasticity c The value s of x for which total revenue is a maximum assume that x is in dollars a Find the equation for elasticity E x b Find the elasticity at the given price stating whether the demand is elastic inelastic or has unit elasticity E 1 Simplify your answer Type an integer or a fraction Is the demand at x 1 elastic inelastic or does it have unit elasticity OA unit elasticity OB inelastic OC elastic c Find the value s of x for which total revenue is a maximum assume that x is in dollars OA Round to the nearest cent as needed Use a comma to separate answers as needed OB The total revenue is independent of x

The average cost for a company to produce x units of a product is given by the function A x The change in average cost is approximately dollars 16x 200 X C Use A x to estimate the change in average cost as production goes from 100 units to

Social animals that live in groups can spend less time scanning for predators than solitary individuals However they waste time fighting with the other group members over the available food There is some group size at which the net benefit is greatest because the animals spend least time on these unproductive activities and thus can spend time on feeding mating etc Assume that for a group of size x the fraction of time spent scanning for predators is 1 x 1 S x A and the fraction of time spent fighting with other animals over food is F x B x 1 where A B are constants and A 2B Find the size of the group for which the time wasted on scanning plus time wasted on fighting is smallest

The frame for a kite is to be made from six pieces of wood The four exterior pieces have been cut with the lengths indicated in the figure To maximize the area of the kite how long should the diagonal pieces be List the diagonal pieces lengths in non decreasing order Your answers may depend on a and b and assume a b a a b b

A rectangle is inscribed with its base on the x axis and its upper corners on the parabola y 11 What are the dimensions of such a rectangle with the greatest possible area Width Height

Find the distance from the origin to the closest point s on the graph of 14x 4xy 14y We begin by computing implicitly for points x y on the graph obtaining dy da We need to minimize the following function of a and y f x y There are four critical points located at the following x values listed in increasing order 21 202 003 1 Thus there is are x y and its distance from the origin is EB E 15 point s closest to the origin one of which is

The frame for a kite is to be made from six pieces of wood The four exterior pieces have been cut with the lengths indicated in the figure maximize the area of the kite how long should the diagonal pieces be List the diagonal pieces lengths in non decreasing order Your answers may depend on a and b and assume a b m b b