Calculus Questions

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

The graph shows the function y w t 1 5 4 3 2 7 1 2 3 4 5 We define a new function a z X 0 1 2 3 1 Complete the table of values for the function a a x 4 2 5 4 15 6 w t dt

Find the derivative of 9e Use e x for e 91 21 cos 2x Hint use product rule and chain rule Now find the equation of the tangent line to the curve at x 0 Write your answer in ma b format

Refer to the figure to the right Set up a definite integral from a to b that represents the shaded area I Ay b y kx

Use the definition of continuity to determine the value of the constant c for which the function fis continuous on where f y cy 8 if y 5 c y 8 if y 5

Find the derivative of the function g x g x e 5x 8 5x 3 2 ez 3 52

Use the Fundamental Theorem of Calculus to find the derivative of 1 f x 1 t5 dt f 5 f x

Evaluate the limit using L H pital s rule e 3x 1 3x lim 0

Suppose that f is a function given as f x follows 1 x 7 We will compute the derivative of fat x 5 as First we compute and simplify the expression f 5 h f 5 h f 5 lim h 0 5 and x 5 h Then we compute and simplify the difference quotient between a f 5 h f 5 h The derivative of the function at x is the limit of the difference quotient as h approaches zero f 5 h f 5 h

2 10 4x dx 1

A factory selling cell phones has a marginal cost function C x 0 01x 3x 229 where a represents the number of cell phones and a marginal revenue function given by R x 429 2x Find the area between the graphs of these curves and x 0 What does this area represent O total revenue for 200 cell phones sold O total profit for more than 200 cell phones sold total profit for 200 cell phones sold O total cost for 200 cell phones sold

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

Find the area between y x 2 and y x 4 Round your limits of integration and answer to 2 decimal places F

Find the area between y 9 and y x 1 with 0 a

Graph the curves r 2 4 cos 0 and r 2 4 sin 0 Identify the correct sketch of r 2 4 cos 0 O A 8 8 OA 8 Identify the correct sketch of r 2 4 sin 0 8 Q 8 U ON OB B 8 8 A 001 Q ON AST OC O C 84 18 T Di Q Q This test 25 point s possible This question 4 point s possib OD OD B C 8 Q

Recall that cot x cos x sin x

Find the area of the region enclosed by f x x and g x V Write an exact answer fraction Answer

Determine the area enclosed by f x x 2x 6x 7 and g x x 3x 8x 17 The region in question lies between x The upper function is g x The area is A and x and the lower function is f x

a the z axis at 1 2 3 b the x axis at 2 6 1 c the y axis at 2 4 4 a Choose the correct equation of a plane perpendicular to the z axis the plane perpendicular to each of the following axes at the given points with a single equation or a pair of O A x 1 B y 2 C Z 3 b Choose the correct equation of a plane perpendicular to the x axis A Z 1 B x 2 O c y 6 c Choose the correct equation of a plane perpendicular to the y axis A y 4 B x 2 www

Find the area between y x 2 and y x 4 Round your limits of integration and answer to 2 decimal places The area between the curves is square units

Find the area between y 9 and y x 1 with x 0 a The area between the curves is square units

Acompany is considering a new manufacturing process It knows that the rate of savings in dollars per year from the process will be about S t 4000 t 4 where t is the number of years the process has been in use Find the total savings during the first year Find the total savings during the first 8 years The total savings during the first year is Simplify your answer The total savings during the first 8 years is Simplify your answer

H1 10 9 7 6 5 4 3 y A x 2 1 y H x 8 3 2 1 27 1 2 1 2 3 4 5 6 7 8 9 10 Write an integral that represents the area of the shaded region shown above Use the function names of th curves given da

Use the given transformation to evaluate the integral f S x y dA where R is the square with vertices 0 0 4 6 6 4 10 2 and x 4u 6v y 6u 4v

6 Estimate the value of 1 n 0 to within 5 decimal places n x n 1 2n 1 when x 0 5 Your estimate should be accurate

Find the present value of a single future payment of FV 35 000 to be made 8 years from today assuming an interest rate of 1 6 per year compounded continuously Identify the correct present value formula for the given values OA PV 35 000 e 0 016 8 OB 35 000 PV e 0 016 8 OC PV 35 000 e 0 016tdt 0 O D PV 35 000 e 1 6 8 4 PV Do not round until the final answer Then round to the nearest hundredth as needed

Evaluate the following indefinite integral J 3 3t 3 dt

It takes a force of 43 764 lb to compress a coll spring assembly from its free height of 11 inches to its fully compressed height of 5 inch a What is the assembly s force constant b How much work does it take to compress the assembly the first half inch the second half inch Answer to the rearest In lb a k lb in b How much work does it take to compress the assembly the first half inch in lb Round to the nearest whole number as needed How much work does it take to compress the assembly the second half inch in b Round in the nearest whole number as needed 301

The marginal revenue in thousands of dollars from the sale of x gadgets is given by the following function The revenue from 120 gadgets is 10 854 2 1 3 33 MR x 4x x 29 000 a Find the revenue function b What is the revenue from selling 250 gadgets c How many gadgets must be sold for a revenue of at least 50 000 a R x Simplify your answer Round to the nearest integer as needed b The revenue from selling 250 gadgets is Type a whole number c To generate a revenue of at least 50 000 approximately Type a whole number gadgets must be sold

Use a graphing calculator to approximate the area between the graphs of y In x and y 3xe on the interval 1 4 The area is ww

sin x Recall that tan x cos x Using integration by substitution to derive the derivative of tan x Edit

S 5 sin 9x 1 dx a This can be transformed into a basic integral by letting Consider the indefinite integral u 9x 1 and du 9dx Jamai b Performing the substitution yields the integral cos 9x 1 C one 9 X syntax error Check your variables you might be using an incom c Once we integrate and substitute the final answer in terms of x is 9 cos 9x 1 C

Use Green s Theorem to evaluate the line integral along the given positively oriented curve Scxydx x y dy C is counterclockwise around the triangle with vertices 0 0 1 0 and 1 4

Evaluate ff fr x y dV where T is the solid bounded by the cylinder x y 1 and the planes z 4 and z 8

Determine if the given polar coordinates represent the same point 8 1 2 8 3x 2 DO Yes No

Determine whether or not the vector field is conservative If it is conservative find a function f such that F Vf F x y 6x 5y i 5x 4y j

Find a vector of magnitude 3 in the direction of v 12i 5k The vector is i 0 Simplify your answer Use integers or fractions for any numbers in the expression

Find the length and direction when defined of u xv and vxu u 6i 4j 6k v 3i 2j 3k OA i k Type exact answers OB The direction of uxv is not defined lux v Simplify your answer What is the direction of uxv Select the correct choice below and if necessary fill in the answer boxes to complete your choce of 6 A i k Type exact answers vxu Simplify your answer What is the direction of vxu Select the correct choice below and if necessary fil in the answer boxes to complete your ch ction of vxu is not defined

Evaluate the line integral 4xyds where C is the line segment joining 4 5 to 5 4

A thin wire is bent into the shape of a semicircle x y 4 x 0 If the linear density is 4 find the exact mass of the wire

Consider the indefinite integral This can be transformed into a basic integral by letting u ln z and du 1 Z In 25 5 ln z dz Z dz Performing the substitution yields the integral In C du syntax error Check your variables you might be using an

Find the work done by the force field F on a particle that moves along the curve C where F x y 5x 5y i 4xyj C r t 3t i t j 0 t 3

Convert this integral from rectangular coordinates to cylindrical coordinates Do not evaluate the integral S 4 2 16 2 So yzdzdydx

Set up the following triple integral but do not calculate it a solid that lies within the cylinder x y 4 and above the xy plane and below the cone z 4x 4y Use cylindrical coordinates

Compute the given integral sin sin r cos xdr

Evaluate the indefinite integral 2 x 14 x4dx

Use the shell method to find the volume of the solid generated by revolving the shaded region about the x axis The volume is This question 1 point s possible 3 Submit y 2 V2 0 X