Calculus Questions

10 Given the series e 2 If the series converges indicate the value n 1 that it converges to If it diverges write diverges

12 Test the series for convergence using the Ratio Test 7 n

6 Evaluate the improper integral St 1 dx 2 x 1

13 Write out the Taylor series that represents the function f x x at the point a 1

5 Evaluate the definite integral e4 dx xvlnx

11 Test the series for convergence using the Root Test 00 n 1 6n 5n 4 n

7 Given the sequence an n 5 If the sequence converges indicate the 6 n value that it converges to If it diverges write diverges

4 Write as Partial Fractions x 6 x 7

Solve 2 sin t 5 sin t 2 0 for all solutions 0 t 2

Solve 8 sin t 2 cos t 5 0 for all solutions 0 t 2

Solve 8 sin t 2 cos t 5 0 for all solutions 0 t 2T

Solve 6 cos t 7 cos t 2 0 for all solutions 0 t 2

Another model for a growth function for a limited population is given by the Gompertz function which is a solution of the differential equation below where c is a constant and K is the carrying capacity K dP dt cln a Solve this differential equation P 0 Po P t c 0 P b Compute t lim P t lim P t If there is no solution enter NONE in the answer box

Evaluate Sc xy y 2 ds along the curve r t 2ti tj 3 2t k 0 t 1 Sc xy y 2 ds Simplify your answer

Evaluate Sc xy x 2 ds along the curve r t 2ti tj 9 2t k 0 ts1 Sc xy x 2 ds Simplify your answer

a Find the solution to the initial value problem y e 6 y lim y t 1 00 6 y 0 1 b Discuss the behavior of the solution y t as t becomes large Does lim y t exist If the limit exists enter its value If the limit does not exist enter 1 00 DNE help numbers help equations

Graph the following function y 3 sec x 7 2 Drag the movable red point to shift the function the black point to set the vertical asymptotes and the blue point whose value is printed on the graph at the correct set of coordinates You may click on a point to verify its coordinates Note Make sure to move the points in the direction of the phase shift represented in the function Provide your answer below 2TT 37 2 TT TR2 8 7 6 5 4 0 1 2 3 4 TR2 3TT 2 2TT

32 e 4 t 3 At least one of the answers above is NOT correct 1 point The unique solution to the initial value problem Yo 6 is y t 8e 31 2 Determine the constant yo and the function g t help numbers g t 32e 41 3 I y y g t y 0 yo 32e 4 3 help formulas

y 1 t 2 4 t 2 4 2 C The answer above is NOT correct 1 point Find the general solution to Enter your answer as y Use C to denote the arbitrary constant in your answer y 1 t 2 4 t 2 4 2 C y help equations 1 22 4 4 y 2ty f 4 4 C

Use the substitution t tan 2 6 sin 2x 2 cos x dx to transform the integrand into a rational function of t and then evaluate the integral

A company manufactures and sells x cellphones per week The weekly price demand and cost equations are given below p 400 0 1x and C x 20 000 140x A What price should the company charge for the phones and how many phones should be produced to maximize the weekly revenue What is the maximum weekly revenue The company should produce 2000 phones each week at a price of 200 Round to the nearest cent as needed The maximum weekly revenue is 400000 Round to the nearest cent as needed B What price should the company charge for the phones and how many phones should be produced to maximize the weekly profit What is the maximum weekly profit The company should produce phones each week at a price of Round to the nearest cent as needed

Express the given quantity as a function f x of one variable x the perimeter of a rectangle of length x and width y that has an area of 187 square meters f x

A fence is to be built to enclose a rectangular area of 1800 square feet The fence along three sides is to be made of material that costs 5 per foot The material for the fourth side costs 15 per foot Find the dimensions of the rectangle that will allow for the most economical fence to be built The short side is 300 ft and the long side is 600 ft

A parcel delivery service will deliver a package only if the length plus girth distance around does not exceed 120 inches A Find the dimensions of a rectangular box with square ends that satisfies the delivery service s restriction and has maximum volume What is the maximum volume B Find the dimensions radius and height of a cylindrical container that meets the delivery service s restriction and has maximum volume What is the maximum volume A The dimensions of the rectangular box are Use a comma to separate answers as needed in COLLE

Apply Green s Theorem to evaluate the integral 8y x C x dx y 2x dy C The circle x 8 y 7 7 8y C x dx y 2x dy

Find the potential function f for the field F 6x i 7y j 9z k 2012 TRONISAT A general expression for the infinitely many potential functions is f x y z

Of all numbers whose difference is 2 find the two that have the minimum product What are the two numbers Use a comma to separate answers as needed

A car rental agency rents 440 cars per day at a rate of 40 per day For each 1 increase in rate 10 fewer cars are rented At what rat should the cars be rented to produce the maximum income What is the maximum income The rental agency will earn a maximum income of when it charges per day

A company manufactures and sells x television sets per month The monthly cost and price demand equations are C x 74 000 70x and p x 300 0 x 9000 X 30 A Find the maximum revenue B Find the maximum profit the production level that will realize the maximum profit and the price the company should charge for each television set C If the government decides to tax the company 5 for each set it produces how many sets should the company manufacture each month to maximize its profit What is the maximum profit What should the company charge for each set A The maximum revenue is Type an integer or a decimal

Find the dimensions of a rectangle with a perimeter of 184 feet that has the maximum area The side lengths are feet Use a comma to separate answers as needed

3x 6 Sketch the following function f x BY HAND Determine the following 2 x x 6 characteristics and communicate your process Make sure to communicate these calculations without having to look at the graph The idea is that you can graph rational functions without technology and have the algebra to prove it x intercept s y intercept vertical asymptote s horizontal asymptote behaviours at the asymptotes It s okay to check Desmos for the sketch of the function but make sure it matches the information you have

Show that the value of the integral below does not depend on the path taken from A to B B dx A dx 2y dy 2xz dz Let M z N 2y and P 2xz Apply the Test for Exactness P N 2xz 2y

Find a potential function for F 3 4x 2 8x F i y y j x y y 0 A potential function for F is f x y z

Find the k component of curl F for the following vector field on the plane F 7y sin x i x sin 3y j curl F k www

Given f x x 2x A Find f x B Find the slopes of the lines tangent to the graph off at x 3 0 and 2 C Graph f and sketch in the tangent lines at x 3 0 and 2

Step 4 Since the square root and the square cancel each other we have 3 L e e d dt Submit 11 3 1 Skin you cannot come back

Now Step 3 dx dt dy et e t 2 te Submit Skip you cannot come back 2t e t 2 e 2t The simplified expression e2t 2 e 2t inside the radical can be re written as et 4

So dx dt Step 2 Now dx dt 4 dy dt et e t and et e 2 2 X

Find the exact length of the curve x et e t Step 1 For a curve given by parametric equations x f t and y g t arc length is given by dx L 2 dt dt We have x et et y 5 2t 0 t 3 So y 52t Osts 3 dx dt and dy 15 Submit Skip you cannot come back dt

in a room where the temperature is 20 C a When do you think the coffee cools most quickly What happens to the rate of cooling as time goes by Explain The coffee cools most quickly as soon as Select b Newton s law of cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings provided that this difference is not too large Write a differential equation that expresses Newton s law of for this particular situation Use t as the independent variable y as the dependent variable R as the room temperature and k as a proportionality constant dy dt What is the initial condition y 0 In view of your answer to part a do you think this differential equation is an appropriate model for cooling Select dy dt approaches Select The answer and the model 974 c Make a rough sketch of the graph of the solution of the initial value problem in part b y 20 because as y approaches R The rate of cooling Select towards Select since the coffee approaches room temperature t DO y 979 so the model seems appropriate o 774 779 20

1 Sketch a graph of the function Then sketch a graph of the inverse of each function Label each graph with its equation Lastly complete the table K a f x 2 x int y int Domain Range Asymptote f x 10 9 8 7 3 2 1 x int y int Domain Range Asymptote 10 9 8 7 6 5 4 3 2 0 4 2 3 4 5 6 7 8 9 10 2 3 f x 4 5 6 8 9 10

a For what values of r does the function y ex satisfy the differential equation 7y 34y 5y 0 Enter your answers as a comma separated list r b If r and r are the values of r that you found in part a show that every member of the family of functions y ae 1 be 2 is also a solution Let r be the larger value and r be the smaller value We need to show that every member of the family of functions y ae 1 be 2 is a solution of the differential equation 7y 34y 5y 0 We must decide whether 7f x 34f x 5f x 0 for f x ae 1 be 2 Substitute your values for r and r into f x then find f x and f x We have f x Substituting and combining like terms we get the following 7f x 34f x 5f x and f x Need Help Therefore f x ae 1 be 2 is a solution to the differential equation 7y 34y 5y 0 Read 11 Watch It Master It

Evaluate f x y ds x y ds where C is the straight line segment x 4t y 8 4t z 0 from 0 8 0 to 8 0 0 f x y ds

Tutorial Exercise Find the exact length of the curve x et e t y 5 2t Ost 3 Click here to begin

Find the line integral of f x y ye along the curve r t 3t i 4tj 2sts 1 The integral of f is Type an exact answer

x y 2 ds where C is parmeterized by Find the value of integral t 8 sin t 2 7 cos t 6 cos t for 0 t

Find the line integrals of F 4yi xj 2zk from 0 0 0 to 1 1 1 over each of the following paths a The straight line path C r t ti tj tk 0st 1 b The curved path C r t ti t j t k Osts1 c The path C3UC4 consisting of the line segment from 0 0 0 to 1 1 0 followed by the segment from 1 1 0 to 1 1 1 0 0 0 C 1 1 1 CA 1 1 0 a The line integral of F over the straight line path C is Type an integer or a simplified fraction

22 e Find the gradient field of the function g x y z 2 In x 3y The gradient field is Vg Oi Oj O k

Consider the following graph 0 The function in the graph is a solution of one of the following differential equations A y 1 xy B y 2xy C y 1 2xy Decide which is the correct equation and justify your answer For points in the first quadrant the differential equation y 1 xy Select graphed Select in the first quadrant and For points in the first quadrant the differential equation y 2xy Select graphed Select in the first quadrant and For points in the first quadrant the differential equation y 1 2xy Select graphed Select in the first quadrant and To summarize the graph is the solution to which of the three differential equations O A y 1 xy OB y 2xy OC y 1 2xy Select Select Select MY NOTES v Since the For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation Since the fur in the second quadrant this differential equation Select be the correct equation Since the For points in the second quadrant this differential equation Select in the second quadrant this differential equation Select be the correct equation For points in the second quadrant this differential equation Select ASK

A tank contains 2 000 L of brine with 16 kg of dissolved salt Pure water enters the tank at a rate of 20 L min The solution is kept thoroughly mixed and drains from the tank at the same rate a How much salt is in the tank after t minutes kg yo b How much salt is in the tarik after 30 minutes Round your answer to one decimal place y kg