Calculus Questions

Problem 26 Suppose 7 R R be a linear transformation whose standar matrix has the basis 0 1 2 1 0 1 2 1 1 Find an orthonormal basis R use the usual inner product on R THE

2 15 pts Simplify the following cosx tanx COSX 1 sinx 3 15pts Evaluate cos 0 3 where cos0 3 4 with in 2th Quadrant and sin 5 7 wi B in Quadrant II 16 Find an equivalent expression for cot t

5 15 pts If tanx 1 6 and x is in the fourth quadrant 6 18 pts Prove the following tan x sin x tan xsin x

Given S is a surface for a part of the sphere x y z 4 that lies above the cone z x y Find a parametric representation for the surface S in term of o and 8 Then determine the domain of and 0 Based on the expression written previously show that rx rell 4 sin o Hence compute the surface integral of ffs ds where the surface S is given parametrically by r 0 0 Let C is the boundary curve along the bottom of the upper half of the sphere thus use Stokes theorem to compute the line integral of fo F dr where F x y z xzi yzj xyk and upwards orientation

Find a power series for the function centered at c 5x x 3x g x I g x C 0 100 n 2 1 1 1 n 0 Determine the interval of convergence Enter your answer using interval notation 1 1

Find the partial fraction decomposition of the rational function x 14 x 2x 8 2 x 2 3 2 2 2

Differentiate f x sin x cot x 4 F x cos x csc 2 x

Use the power series 1 1 x to find a power h x h x 00 n 0 00 1 x Ixl 1 n 0 series for the function centered at 0 2 x 1 x 1 1 1 1 x 1 X Determine the interval of convergence Enter your answer using interval notation 1 1

Find each equilibrium solution x t x of the given second order differential equation x f x x 0 Use a computer or graphing calculator to construct a phase portrait and direction field for the equivalent first order system x f x y Thereby ascertain whether the critical point 0 looks like a center a saddle point or a spiral point of t system x x x 0 Comp The equilibrium solution s is are x t Type an integer or a fraction Use a comme to separate answers as needed

3 The asymptotic expression f n O g n means there exists a constant c 0 and an integer No such that for all nz No f n sc g n Consider f n n and g n gn It is a fact that n O lg Determine the smallest positive constant No such that for all n No f n sc g n given that c 32 Be careful The floor over the log makes this problem interesting 5 points

Find the radius of convergence of the power series If you need to use co or co enter INFINITY or INFINITY respectively 1 n 0 n 9

Find the interval of convergence of the power series Be sure to include a check for convergence at the endpoints of the interval If the answer is an interval enter your answer using interval notation If the answer is a finite set of values enter your answers as a comma separated list of values 1 4x 1 n 1 n 1 11 11

ind the interval of convergence of the power series Be sure to include a check for convergence at the endpoints of the interval If the answer is an interval enter your answer using interval notation If the answer is a finite set of values enter nswers as a comma separated list of values 1 x 2 n8n n 1 6 10

State where the power series is centered nx n 0

sinx cos2x 1 Write your answer in radians in

Given a generic point M in the Argand diagram representing the complex number z and let 8 iz 61 zi s t Re U 0 Determined the circle representing the locus of M Find the locus of M whenever the complex number U is real

Use the figure to evaluate the definite integrals g 1 dt and 2 1 2 8 1 a 1 g t dt y g t 2 3 4 5 8 Express numbers in exact form Use symbolic notation and fractions where needed g t dt

The function f x 16x 4 is one to one a Find the inverse of f and check the answer b Find the domain and the range of f and f c Graph f f1 and y x on the same coordinate axes

Find the equation of the graph given below using the points provided Notice that the cosine function is used in the answer template representing a cosine function that is shifted and or reflected Use the variable x in your equation but be careful not use the multiplication x symbol 34 2 Provide your answer below N TT TT 2 2 3 0 1 2 3 6 0 TT 2 E 3 t

graph be the two points that are labeled on the graph given below Notice that the sine unction is used in the answer template representing a sine function that is shifted and or reflected Use the variable x in your equation but be careful not use the multiplication x symbol 2 1 0 1 2 3 0 4 Provide your answer below TT 2 57 TT 3TT 2 3T 2 2TT

ind and sketch the level curves f x y c on the same set of coordinate axes for the given values of c Refer to these level curves as contour map z 3x y c 2 1 0 1 2 Choose the correct graph below A O 2 OB 2 2 Q Q 5 OC 1 Q 5 OD 2 O OU

f f 3 For the function f x y sin 5x 8y find and dx dy f x f 0

dz Draw a dependency diagram and write a chain rule formula for where z h m n m g q n f q dq Choose the correct dependency diagram below OA dn n z h N OB dz dq Choose the correct chain rule formula A dz dq OC dz dq OD dz 11 dz dh dm m az dm az dn am dq on dq dz dm dz dn am dq an dq dz dm dz dn am dq an dq az dn az dm I B m z h m n dz dm dm dq 9 Uz on dn B n C m q dz dm dm dq z h m n m m n dq dz n q

Sketch the graph described by the following cylindrical coordinates in three dimensional space r 6 Choose the correct answer below A 12 OE 12 12 12 OB 11 6 OF 126 H 12 12 X 12 X 12 O C 12 0 O G X 12 OD OH X

Sketch the region of integration associated with the double integral S S f x y 1 X Choose the correct sketch of the described region OA f x y dydx N 00 M OB 2 15 OC 5 Q 27 OD 5

a x y Find the image d u v Solve the system u 2x y v 3x 2y for x and y in terms of u and v Then find the value of the Jacobian under the transformation of the triangular region with vertices 0 0 2 4 and 2 3 in the xy plane Sketch the transformed region the uv plane The function for x in terms of u and vis x The function for y in terms of u and v is y The Jacobian of the transformation is J u v Choose the correct sketch of the transformed region in the uv plane below OA OB OC 0 14 OU 7 0 0 14 ELTER Q Q 0 14 7 0 D 0 14 7 0 ON

Evaluate the iterated integral 2 1 25 S S 18x y dy dx 21 5 18x y dy 1 0 dx x Simpaty Simplify your answer

Evaluate the integral 2 X x 4x 20 dx

Use integration by parts together with the techniques of this section to evaluate the integral Use C for the constant of integration x tan tan x dx

Evaluate the integral x x 1 x 1 x 2 dx

Evaluate the integral 4y 5y 12 dy y y 2 y 3

Evaluate the integral L z X 4x 20 cx

On the graph of f x sec x and the interval 2 0 for what value of x does f x 1 Choose all answers that apply Select all that apply 2 0 37 IT 32

Given that sec x 2 what is sec x

a Write out the form of the partial fraction decomposition of the function as in this example Do not determine the numerical values of the coefficients x6 x2 4 b x x 1 x 8

Problem 6 Sketch a graph of the function f that satisfies the following conditions f is continuous on f 2 1 f 0 0 f 2 1 f 0 0 f x 0 on 0 f x 0 on 0 f 2 0 f 2 0 f x 0 on 2 and 2 00 f x 0 on 2 2 lim f x 2 lim f x 2 8118 218 Solution

6 A patient is injected with a drug and t hours later the concentration of the drug remaining in the patient s mg cm What is the average concentration of drug during the 3t bloodstream is given by C t 1 36 first 8 hours after the injection 10pts Use substitution

On the graph of f x csc x and the interval 2 0 set the moving point to a value of x such that f x 1 Provide your answer below 2 T 3TT 2 I 1 1 1 I 13 10 0 1 TT 2 1 I 10 5 0 5

Evaluate the integral 2 6 2x 3x 3 dx 1

On the graph of f x csc x and the interval 0 2 set the moving point to a value of x such that f x Provide your answer below 5 TT 2 1 11 11 1 TT 11 11 11 10 0 11 11 3TT 2 11 JE 11 THE 2TT 11 11 11 5TT 2

The equation T 0 6sin 2 t 8 101 4 describes the body temperature of a cow in Fahrenheit where t is time in 2 Do not include the units in your hours What is the temperature of the cow to the nearest tenth of a degree when t answer NOTE The angle is in radians

a b c d dx X 7 S Since the function y Since the function y Since the function y Since the integral dx x 9 Since the integral Since the function y Since the function y Since the integral 8 S Since the function y Since the integral tan x dx 8 5 Since the integral Since the integral ex dx X Since the integral 10 Since the integral 1 Since the function y 1 Since the function y Since the function y 1 Since the function y tan x has an infinite discontinuity at x 0 the integral is a Type 2 improper integral Since the function y tan x has an infinite discontinuity at x the integral is a Type 2 improper integral 2 Since the function y tan x has an infinite discontinuity at x 1 the integral is a Type 2 improper integral 1 X 7 00 dx 1 has an infinite discontinuity at x 1 the integral is a Type 2 improper integral x 7 1 has an infinite discontinuity at x 7 the integral is a Type 2 improper integral X 7 X 00 dx has an infinite interval of integration it is a Type 1 improper integral x 9 has an infinite interval of integration it is a Type 2 improper integral x 9 dx 1 dx X 7 1 1 has an infinite discontinuity at x 9 the integral is a Type 1 improper integral x 9 1 has an infinite discontinuity at x 8 the integral is a Type 1 improper integral x 9 1 has an infinite discontinuity at x 7 the integral is a Type 1 improper integral x2 has an infinite discontinuity at x 9 the integral is a Type 2 improper integral 9 has an infinite interval of integration it is a Type 1 improper integral has an infinite interval of integration it is a Type 2 improper integral tan x dx has an infinite interval of integration it is a Type 1 improper integral tan x dx has an infinite interval of integration it is a Type 2 improper integral X X ex has an infinite discontinuity at x 1 the integral is a Type 1 improper integral X has an infinite discontinuity at x 0 the integral is a Type 1 improper integral ex has an infinite discontinuity at x 0 the integral is a Type 2 improper integral X X ex dx has an infinite interval of integration it is a Type 1 improper integral ex dx has an infinite interval of integration it is a Type 2 improper integral

c tan x dx 11 Since the function y tan x has an infinite discontinuity at x 0 the integral is a Type 2 improper integral Since the function y tan zx has an infinite discontinuity at x the integral is a Type 2 improper integral Since the function y tan x has an infinite discontinuity at x 1 the integral is a Type 2 improper integral O Since the integral tan x dx has an infinite interval of integration it is a Type 1 improper integral tan x dx has an infinite interval of integration it is a Type 2 improper integral Since the integral

What is the amplitude and period of the function shown below 3 Provide your answer below 2 1 1 0 1 24 3 4 5 56 6 0 2 3 4 5 6 8

Given that csc x 9 29 what is csc x

Explain why each of the following integrals is improper s dx x 7 O Since the function y O Since the function y O Since the integral 1 O Since the function y O Since the integral dx x 9 O Since the function y O Since the function y O Since the function y 500 1 x 7 has an infinite discontinuity at x7 the integral is a Type 1 improper integral O Since the integral O Since the integral I 1 x 7 1 X dx dx x 7 has an infinite discontinuity at x 1 the integral is a Type 2 improper integral has an infinite discontinuity at x 7 the integral is a Type 2 improper integral has an infinite interval of integration it is a Type 1 improper integral has an infinite interval of integration it is a Type 2 improper integral 1 has an infinite discontinuity at x 9 the integral is a Type 1 improper integral x 9 1 x 9 1 9 has an infinite discontinuity at x 9 the integral is a Type 2 improper integral dx x 9 dx 9 has an infinite discontinuity at x 8 the integral is a Type 1 improper integral has an infinite interval of integration it is a Type 1 improper integral has an infinite interval of integration it is a Type 2 improper integral

a Sketch the garden use INSERT SHAPES and INSERT DRAW b How many meters of stone do you need to create the border FO 12 9 Danielle wants to create a patio in her backyard She would like it to be 6 ft by 12 ft What is the area of the patio Sketch the patio INSERT SHAPES and INSERT DRAW C 13 10 A basketball with a diameter 24 cm is packaged so that it just fits into a cube shaped box Calculate the amount of empty space in the box round to 1 decimal place T 17

American General offers a 18 year annuity with a guaranteed rate of 6 47 compounded annually How much sho you pay for one of these annuities if you want to receive payments of 1300 annually over the 18 year period How much should a customer pay for this annuity Round to the nearest cent

z and z where z 9xy 3x y x 3s 3t and y 3s 3t 4 0 Type an expression usings and t as the variables 4 0 Type an expression using s and t as the variables

On the domain of 0 2 set the moving point to a value of x such that cos x sec x Provide your answer below 10 5 0 5 TT 3 27 3 y 13 I 1 1 T 4 5 0 TT 4TH 3 5TT 3