# Recently Submitted Homework Problems: calculus

23. T.J., the owner/operator of the Commonplace Coffeehouse here in Indiana has determined that the daily demand (in people) for a medium-sized cappuccino made with Costa Rican coffee beans is given by the equation: D(p) = 325e (negative 0.4p) Where p is the price of the coffee drink in dollars. Calculate E(p), the elasticity of demand.(Answered)117. What is the difference b/w relation and function. It would be great if you could give some real life example.

(Answered)

348. Find the critical numbers of f (x) = x^2-6x. Find also the open intervals on which the function is increasing or decreasing and locate all relative extrema.(Answered)

353. work out the grap y =x^-3(Answered)

634. indefinite integral of sqrt(3-2x)dx(Answered)

2863. sin5x dx + 2y cos^3 (5x) dy =0(Answered)

1453. Help me understand how xe^2 - e^x = e^2 - e^x(Answered)

738. given e^x solve for x

(Answered)

1823. After takeoff, an airplane climbs at an angle of 35° at a speed of 180 ft/sec. How long does it take for the airplane to reach an altitude of 14,000 ft? (Round your answer to one decimal place.)

min(Answered)

771. Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers \"c\" that satisfy the conclusion of the Mean Value Theorem.

f(x)=3x^2 + 2x + 5(Answered)

806. Show that the circumference of the unit circle is equal to 2*2* (Integral [-1,1] SqRoot (1-x^2)] dx)(Answered)

960. A 75 mg dose of penicillin is given to a patient to prevent malaria. Penicillin leaves the body at a rate of

6.5% per hour.

a. Find a formula for the amount A(in mg) of penicillin in the body t hours after the dose is given.

(Answered)

1029. A triangle ABC has sides of length AB = 3 inches and AC = 4 inches. Let t denote the angle at vertex A, and let s denote the length of the remaining side BC. If t increases at the rate of 2 radians per second, at what rate (in inches per second) does s change at the instant when t is a right angle? [Hint: You will need to use the Law of Cosines.](Answered)

1051. a right triangle is to be constructed as that the sum of the lengths of the legs x=ab and y=bc is 10 inches. what should x and y be to minimize the length of the hypotenuse ac?(Answered)

1134. Integration by parts

e^4*x*cos(4*x)dx

(Answered)

1136. Help me solve this math problem.

Let R be the region bounded by x=1, x=5, the axis and y=1/x. Approximate the area of R using 8 right rectangles.(Answered)

1143. find the rate of change of the distance between the origin and a moving point on the graph of y=x squared + 1 if dx/dt = 2 centimeters per second(Answered)

1146. If (a sub n) is convergent and (a sub n) does not equal 0, prove that (1/(a sub n)) is divergent.(Answered)

1245. Write the Maclaurin series for each function:

a)e^(x/2)

b) sin(3x)

c)((x^2)/2)-(1)+(cosx)(Answered)

1186. The sum of two positive numbers is 20. Find two numbers such that: a) THe sum of the squares is a minimum. b) the product of one and the square of another is a maximum. c) the product of the square of one adn the cube of the other is a maximum.

(Answered)

1205. Water is pumped into an empty tank at a rate of r(t)=20e^0.002t gallons per minute. Approximately how many gallons of water have been pumped into the tank in the first five minutes?(Answered)

1230. Suppose that the utility function for two commodities is given by

U = x2y

and that the budget constraint is

x + 2y = 30.

What values of x and y will maximize utility?

(x, y) = ( )(Answered)

1249. A man has $235,000 invested in three properties. One earns 12%, one 10% and one 8%. His annual income from the properties is $22,500 and the amount invested at 8% is twice that invested at 12%.

(a) How much is invested in each property?

12% $ property 10% $ property

8% $ property

(b) What is the annual income from each property?

12% $ property

10% $ property

8% $ property

(Answered)

1251. Use row operations on augmented matrices to solve the given systems of linear equations.

x + y - z = 0

x + 2y + 2z =-5

3x - y - 14z= 19

(x, y, z) = ( )

(Answered)

1252. The system of equations may have unique solutions, an infinite number of solutions, or no solution. Use matrices to find the general solution of the system, if a solution exists. (Enter your answers as a comma-separated list. If there are an infinite number of solutions, enter INFINITY. If there is no solution, enter NO SOLUTION.)

2x+y-z=-16

x-y+2z=27

x+y-z=-1

(x, y, z)

= ( )(Answered)

1254. Write an equation fot the tangent line of the curve -2y^3+5xy=-2x+12 at the point where y=1.(Answered)

1265. Suppose that you have figured out that the demand for the book is given by the function D(t)=sqrt(25-t) where t is the number of days since the zombies have started to surround your fortified parking garage and D(t) is the price per book that the survivors are willing to pay. You have also had time to do a little reconnaissance and have determined that the bookstore has found the supply of books available to be given by the function S(t)= 3t/16 where t is as above and S(t) is the price at which the bookstore is willing to sell the book. (a) Find the number of days t when both the surviving population and the bookstore em- ployees can agree on a price. What is the price of the book at this time? (b) What is the consumer surplus at this equilibrium point? (c) What is the producer surplus at this equilibrium point?(Answered)

1271. A trust account manager has $480,000 to be invested. The investment choices, x, y, and z, have current yields of 8%, 7%, and 10%. Suppose that the investment goal is to earn interest of $37,200, and risk factors make it prudent to invest some money in all three investments.

(a) Find a general solution for the amounts invested at the three rates. (Do not use commas.)

x = $

y = $

z = z

(b) If $9,000 is invested at 10%, how much will be invested at each of the other rates?

x = $

y = $

What if $29,000 is invested at 10%?

x = $

y = $

(c) What is the minimum amount that will be invested at 7%?

y = $

In this case how much will be invested at the other rates?

x = $

z = $

(d) What is the maximum amount that will be invested at 7%?

y = $

In this case how much will be invested at the other rates?

x = $

z = $ (Answered)

1292. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?(Answered)

1294. 1. Given the Price Demand equation p+0.01x = 50

(a) Express the Demand x as a function of the price p.

(b) Find the elasticity of demand, E(p).

(c) What should the company charge for the camera, and how many cameras should be produced to realize the maximum weekly profit?

(Answered)

1295. Set up the system of equations and then solve it by using inverse matrices.

Medication A is given every 4 hours and medication B is given twice per day, and the ratio of the dosage of A to the dosage of B is always 5 to 8.

(a) For patient I, the total intake of the two medications is 23.2 mg per day. Find the dosage of each administration of each medication for patient I. (Round your answers to one decimal place.)

A = mg

B = mg

(b) For patient II, the total intake of the two medications is 87.8 mg per day. Find the dosage of each administration of each medication for patient II. (Round your answers to one decimal place.)

A = mg

B = mg(Answered)

1352. How do you change an equation that has respect to x to have respect to y?(Answered)

1385. let R be the region bounded by the following curves. y=2x, y=0, x=3 Use the disk method to find the volume of the solid generated when R is revolved about the x axis(Answered)

1408. The half-life of the radium isotope Ra 226 is approximately 1,599 years. What percent of a given amount remains after 100 years?(Answered)

1419. I need help with this calculus problem:

Find the derivative of f(x)=x^3/2(Answered)

1419. I need help with this calculus problem:

Find the derivative of f(x)=x^3/2(Answered)

1434. Let R be the region bounded by the x-axis, the y-axis, the line y=2, and the curve y=sqrt(x-1). A container is built in the shape of the solid of revolution formed by rotating R about the y-axis, and the container is filled with cold pressed olive oil. Find the work done in pumping all of the olive oil to the top of the container. (Units meters, density of cold pressed olive oil = 915kg/m^3, acceleration due to gravity is 9.8 m/s^2)(Answered)

1455. Implicitly differentiate the equation to find dy/dx

x^3+y^3= 2x^2y^2+1(Answered)

1467. Help me solve this math problem.... derive tan(a-b) from tan(a+b)(Answered)

1536. evaluate arccot(1/sqrt3)(Answered)

1575. Air is being pumped into a spherical balloon at a rate of 5cm^3/min. Determine the rate at which the radius of the balloon is increasing when the diameter is 18cm.(Answered)

1606. As you know, when a course ends, students start to forget the material they have learned. One model (called the Ebbinghaus model) assumes that the rate at which a student forgets material is proportional to the difference between the material currently remembered and some positive constant,a.

A. Let y=f(t) be the fraction of the original material remembered t weeks after the course has ended. Set up a differential equation for y, using k as any constant of proportionality you may need (let k>0). Your equation will contain two constants; the constant a (also positive) is less than y for all t.

dy/dt=?

y(0)=?

y=?

What are the practical meaning (in terms of the amount remembered) of the constants in the solution y=f(t)? If after one week the student remembers 80 percent of the material learned in the semester, and after two weeks remembers 69 percent, how much will she or he remember after summer vacation (about 14 weeks)?

percent =(Answered)

1641. help me with this calculus problem

The function represents the rate of flow of money in dollars per year. Assume 10 year period and find the present value. f(x)=500e^(.04x) at 13% percent compounded continuously

(Answered)

1643. write a general linear equation for the line that passes through the points (1,1) and (2,1).(Answered)

1648. Integral of (3x^.5+x^-.5)dx(Answered)

1648. Integral of (3x^.5+x^-.5)dx(Answered)

1713. let f(x)=(2x-pi) ^3 +2x-cosx. the value of modulus of d/dx of f inverse of x when x= pi is(Answered)

1722. X^3 - x^2 lessthan equal to 1(Answered)

1741. Prove that the equation has at least one real root and use your calculator to find an interval of length .01 that contains a root.

E^x=2-x

(Answered)

1761. (x-3)e^x=0(Answered)

1763. Determine whether the function f(x)=-x^2-4x+4 has a maximum and minimum

(Answered)

1775. Solve the following DE using Laplace transform.

d^2y/dt2 - 4 dy/dt + 3y = 8e^-t, t > 1(Answered)

1828. Differential Equations: Find, using the technique of separation of variable, a formula(called implicit general solution) satisfied by all solutions of y(x) of this equation: dy/dx=xy/(1+y^4).(Answered)

1850. (2+x)^2-8/x(Answered)

1859. derivative of pi over x^pi: pi/x^pi(Answered)

1945. Help with Calulus

Find the slope of the tangent line to the graph of y=lnx^2 at x=e^2(Answered)

1947. If x^2 y^2=2, find an expression for y.(Answered)

1972. lim (3x+2)/(4x) as x approaches infinity(Answered)

1982. Lim approaches infinity. The square root of x+9-3, all divided by x(Answered)

1985. how to prove that the integral (lnx)^n [eval @ 0 to 1] equal (-1)^n*!(Answered)

1994. An equilateral triangle with sides 1 m is submerged vertically in water so that the base is even with the surface. Express the hydrostatic force against one side of the plate as an integral and evaluate it. (Give your answer correct to the nearest whole number.)(Answered)

2017. differentiate y=2x/1-3x^2(Answered)

2054. find derivative for the function xe^y = x - y(Answered)

2069. differentiate each function

f(x) x3is a exponent-2x+1(Answered)

2091. If the line 3x-4y=0 is tangent in the first quadrant to the curve y=x^3+k, then k is(Answered)

2092. The derivitive of (x)times the square root of (x^2 +1)(Answered)

2097. My question is Evaluate limit of x as it approaches 2,x^1560-2^1560 divided by x-2, using the defintion of the derivative for an appropriate function(Answered)

2098. find dy/dx of x^3+y^3+12xy-8 by implicit differentiation(Answered)

2111. d/dx (2arcsin(x-1)

(Answered)

2111. d/dx (2arcsin(x-1)

(Answered)

2133. Help me understand this Calculus problem: When an object is placed at a distance p

from a convex lens having focal length 6 cms, then the image will be at a distance q from the lens while

1/6=1/p+1/q.

Find the rate of change of p with respect to q.(Answered)

2133. Help me understand this Calculus problem: When an object is placed at a distance p

from a convex lens having focal length 6 cms, then the image will be at a distance q from the lens while

1/6=1/p+1/q.

Find the rate of change of p with respect to q.(Answered)

2170. Find the equation of the line to the curve x^3+x^2y+y^2x=y+2 at point (1,1)(Answered)

2171. find dy/dx if: y=x^(x^1/2)(Answered)

2171. find dy/dx if: y=x^(x^1/2)(Answered)

2178. Help me solve this problem. Find the number of units x that produces the minimum average cost per unit C in the given equation.

C = 0.06x3 + 59x2 + 1384(Answered)

2260. (2x-3)^4(1-x)^3(Answered)

2278. what are the inflection points of

f(x)=(x^4-x^2+2)/x^4 ?(Answered)

2368. Applications of differentials: A poster is to have an area of 180 in^2 with a 1 inch margin at the bottom and either side with a 2 inch margin at the top. What dimensions will give the largest printed area?(Answered)

2382. A particle moving along the curve 16x^2+9y^2=144 find all points which the rate of change of y and x are equal. Assure these rates are never both zero at the same time(Answered)

2409. Help me find all open intervals on which the function f(x) = x to the second power over x to the second power plus 4(Answered)

2415. help me use implicit differentiation on this math problem 2(x+y)^(1/3)=y(Answered)

2551. Find the area enclosed by the x-axis and the function:

f(x)= (x-1) (x+1) (x-3)

(Answered)

2565. Find the maximum value of f(x,y)using the method of the Lagrange Multipliers if the constraint of the function is x+y=70

(Answered)

2720. y=x2 how to find tangent line with points (-4,16)(Answered)

2734. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F... F(t)= 82 + 4sin(t/2) for 0 < t < 30... A) to the nearest whole number, how many cars pass through the intersection over the 30 min period?

b) is the traffic flow increasing or decreasing at t=7

c) what is the average value of the traffic flow over the time interval 10 < t < 15?

d) what is the avg rate of change of the traffic flow over the time interval 10 < t <15?

(Answered)

2820. You invest 5000 in an account which pays interest compounded continuously.

If you want the account to contain 8000 after 6 years, what yearly interest rate is needed?

Round your answer to one decimal place.

(Answered)

2829. Find the lim as x aproaches delta (D) x of: (x+Dx)^2-2(X+Dx)-(X^2-2x)?(Answered)

2923. The Oliver Company plans to market a new product. Based on its market studies, Oliver estimates that it can sell up to 4,600 units in 2005. The selling price will be $2 per unit. Variable costs are estimated to be 20% of total revenue. Fixed costs are estimated to be $6,600 for 2005. How many units should the company sell to break even?(Answered)

2929. The owner of a restaurant plans to price hamburgers so that he makes a profit of $3.25 on each one sold. If he hopes to make a weekly profit of $900 from the sale of the burgers what is the least number he must sell per week(Answered)

2938. (inverse sin) sin power -1 (-1/2) exact value(Answered)

3029. Solve the separable differential equation for

dy/dt = 6y^5

and find the particular solution satisfying the initial condition y(0)=2(Answered)

3031. Find the horizontal and vertical asymptotes of f(x)=(4x^2 -1)/(2x^2+5x-3)(Answered)

3060. Intergreation

determine

( 8 dx

determine

( 2x dx(Answered)

3073. derivative of 2t(3t+1)^3(9t^2+1)(Answered)

3164. A man 6 feet tall walks at a rate of 5 feet per second toward a light that is 20 feet above the ground. When he is 10 feet from the base of the light,

a) at what rate is the tip of his shadow moving?

b) at what rate is the length of his shadow changing?

(Answered)

3215. help me solve this math problem: determine whethe the mean value theorem can be applied to the function f(x)=2sinx+sin2x on the closed interval [8pi,9pi]. (Answered)

3223. if a body falls in vacuum due to action of gravity ,then its acceleration is constant equal to 9.8 m/sec^2.state this law as differential equation for y(t),the distance fallen is taken as function of time and solve to get familiar law y(t)=1/2*g*t^2(Answered)

3249. The product of two positive numbers is 588. Minimize the sum of the first and three times the second.(Answered)

3258. g(t)=t^2(3t+1)^4

find first and second derivatives(Answered)

3329. 1. find dy/dx of x^3+y^3=3xy

2. set dy/dx = 0 and see what happens.

3. use results from step 2 and the original equation to find the coordinates of the point.(Answered)

3360. A particle moves up and down the y-axis with velocity v feet per second given by v=t3 -7t2 +15t - 9 during the time interval [0,4]. At time t=0, its position is y=4.

At what time(s) is the particle stopped?

At what time is is the velocity the absolute maximum? The absolute minimum?

At what time(s) does the velocity-time graph have a point of inflection>

What is happening to the particle at the point(s) of inflection?

Find the position, y, as a function of time.

At what time is the position the absolute maximum? The absolute minimum?

At what time(s) does the position-time graph have a point of inflection?

What is happening to the particle at the point(s) on inflection?

Is y ever negative? (explain)

What is the net displacement of the particle from t=0 to t=4?

How far does the particle travel from t=0 to t=4?

What is the average velocity from t=0 to t=4?

What is the average speed from t=0 to t=4?(Answered)

3377. Suppose that f(x) is an exponential function. Given that f(0)=100 and f(0)=5, what is f(3)?(Answered)

3379. Test for convergence or divergence using the geometric series test for the sum of 5(-1/5)^n from n=0 to infinity.(Answered)

3379. Test for convergence or divergence using the geometric series test for the sum of 5(-1/5)^n from n=0 to infinity.(Answered)

3542. Line L is tangent to y=ln(x) at point P and passes through the point (0,0). Region R is bounded by the graphs of y=ln(x), line L, and the x-axis. What is the equation of L?(Answered)

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