Vector Calculus Questions and Answers

Rewrite the expression as a single logarithm and simplify the result Hint Begin by using the properties of logarithms 7 In cot ti 7 In 1 tan t

e the identity tanh x tanh x 1 tanh x e2x 1 tanh x 1 11 11 cosh x cosh x 2x cosh x e x cosh x 1 ex x ex e x 1 ex e x

Prove the identity cosh x sinh x e First use the definitions of the hyperbolic functions and then simplify by combining the exponential terr e 1 cosh x sinh x ex e x 12 1 et

2 Given the function defined by f x x x 4x 4 Find the zeros off a b Write an equation of the line tangent to the graph off at x 1 The point a b is on the graph off and the line tangent to the graph at a b passes through the point 0 8 which is not on the graph off Find the values of a c and b

Solve the equation in the complex number system x 10x 50 0 What is the solution set Use a comma to separate answers as needed

Write the expression in the standard form a bi 5 3i 5 7i 5 3i 5 7 Simplify your answer

4 1 Points DETAILS LARPRECALCRMRP7 5 1 036 Use the fundamental identities to simplify the expression Use the table feature of a graphing utility to check your result numerically cos x 1 tan x Need Help Read It Submit Answer 5 1 Points DETAILS Factor the expression and use the fundamental identities to simplify Use a graphing utility to check your result graphically sec x tan x sec x STEP 1 Multiply constants 7 sec x tan x sec x STEP 2 Factor LARPRECALCRMRP7 5 1 046 SBS sec x tan x sec x Need Help Read sec x tan x sec x 7 16 STEP 3 Apply the appropriate fundamental trigonometric identity and simplify sec x tan x 6 sec x 16 tan x 1

cot 8 Need Help Submit Answer 3 1 Points DETAILS sec x 1 cot x 01 Read It Match the trigonometric expression with its simplified form tan x csc x O tan x O sin x Need Help Submit Answer Watch it Read It LARPRECALCRMRP7 5 1 022

Current Attempt in Progress Use integration by substitution and the Fundamental Theorem to evaluate the definite integral 13x x 1 dx Enter the exact answer 13x x 13x x 1 dx

Find all values of x satisfying the following conditions x 5 x 9 2 Y23 and y y 1 3 Select the correct choice below and if necessary fill in the answer box to complete your choice A The value s of x for which y y 1 is are Type an integer or a simplified fraction Use a comma to separate answers as needed OB Y y 1 for all values of x There are no values of x for which y y 1

A water tank filtration plant is built in the shape of an inverted cone with height 5 2 m and diameter 5 m at the top Water is being pumped into the tank at a rate of 3 1 2 m min find the rate at which the water level is rising when there is 8 m of water V h 7 marks 3

Solve the system with the addition method 7x 5y 10x 9y Answer x y 33 49 Enter your answers as integers or as reduced fraction s

d The negative angle closest to zero that satisfies cos 0 3 7 is

QUESTION 24 Use the given zero to find the remaining zeros of the function f x x 3x 5x 39 zero 3 3 2i 3 2i 1 2i 1 2i 1 2 131 1 2 13i 6 4i 6 4i QUESTION 25

j f x x 4 x 5 f x h f x h x h 4 x 4 x 5 x h 5 h x 5 x h 4 x 4 x h 5 x 5 x h 5 h

Construct a function that passes through the origin with a constant slope of 1 with removable discontinuities at x 6 and x 4 Enclose numerators and denominators in parentheses For example a b 1 n f x LA sin a 00 E

p f x sin x write as a factor of sin x f x h f x sin x h sin x h h use sin A B sinAcosB sinBcosA sin x cos h 1 sin h cos x h

The function f x 8x 2 e critical number X 41 has one critical number Algebrically use the derivative to find the

a Use the appropriate formula to find the value of the annuity b Find the interest Periodic Deposit 90 at the end of every six months iClick the icon to view some finance formulas Rate Time 6 5 compounded semiannually 40 years a The value of the annuity is Do not round until the final answer Then round to the nearest dollar as needed b The interest is Use the answer from part a to find this answer Round to the nearest dollar as needed x m W SO ty is hal a rt a www Formulas In the provided formulas P is the deposit made at the end of each compounding period r is the annual interest rate of the annuity in decimall is the number of compounding periods per year and A is the value of the after t years P 1 0 1 P A 1 A A C P A

To offer scholarships to children of employees a company invests 15 000 at the end of every three months in an annuity that pays 10 compounded quarterly a How much will the company have in scholarship funds at the end of ten years b Find the interest Click the icon to view some finance formulas a The company will have in scholarship funds Do not round until the final answer Then round to the nearest dollar as needed b The interest is S Use the answer from part a to find this answer Round to the nearest dollar as needed Formulas In the following formulas P is the deposit made at the end of each compounding period r is the annual interest rate of the annuity in decimal form n is the number of compounding periods per year and A is the value of the annuity after t years A P 1 0 1 r A P A P 1 r 8 In the following formulas P is the principal amount deposited into an account ris the annual interest rate in decimal form n is the number of compounding periods per year and A is the future value of the account after t years A n nt nt X

Find the value of the annuity and the interest Round to the nearest dollar A nt A P 1 r 1 r A P 9 A nt OA 21 047 3 047 OB 22 420 4 420 C 29 722 11 722 D 62 420 44 420 1 P Periodic Deposit 900 at the end of every six months Rate 4 5 compounded semiannually Time 10 years

65 A family of curves has polar equations 1 a cos 0 1 a cos 0 Investigate how the graph changes as the number a changes In particular you should identify the transitional values of a for which the basic shape of the curve changes

Use l Hopital s Rule to evaluate the limit cos 3x 1 17 lim X 8 x2 A 96 B 0 6 3 2 D

Factor the polynomial completely Q x x 256 Q x Find all its zeros State the multiplicity of each zero Order your answers from smallest to largest real followed by complex answers ordered smallest to largest real part then smallest to largest imaginary par with multiplicity X X X X with multiplicity with multiplicity with multiplicity

Factor the polynomial completely P x x5 2x3 P x Find all its zeros State the multiplicity of each zero Order your answers from smallest to largest real followed by complex answers ordered smallest to largest real part then smallest to largest imaginary with multiplicity X X X with multiplicity with multiplicity

1 Points Find all zeros of the polynomial Enter your answers as a comma separated list Enter all answers P x x 5x 35x 5x 36 X

So far we have worked only with polynomials that have real coefficients This exercise involves polynomials with real and imaginary coefficients Find all solutions of the equations Enter your answers as comma separated lists Enter all answers including repetitions a 2x 18i 1 X b X c X d X x 4ix 0 x 6ix9 0 ix 2x i 0

A polynomial P is given P x x 15x 16 a Factor P into linear and irreducible quadratic factors with real coefficients P x b Factor P completely into linear factors with complex coefficients P x

Find a polynomial with integer coefficients that satisfies the given conditions Q has degree 3 and zeros 3 2i and 21 Q x

In the process of finding the inverse of the following matrix explain the indicated row operation 3 3 6 1 0 0 1 3 10 0 10 1 3 5 0 0 1 Step 1 Switch row 1 and row 2 1 3 1 3 Answer to Step 1 Step 2 Explain the row operation to change the answer from Step 1 to the result below 1 3 10 0 1 0 6 24 1 3 0 3 5 0 Answer to Step 2 0 3 10 0 1 3 6 100 5 0 0 1 1 0 1

Prove the identity sin x 7 cos x 2 Use the Subtraction Formula for Sine and then simplify sin x sin x cos cos x CO sin x 0 cos x

Verify the identity sin t cos t 2 sec t csc t sin t cos t sin t cos t sin t cos t sin t 2 sin t cos t sin t cos t sin t cos t sin t cos t 1 sin t cos t 2 sec t cs t sin t cos t

Sketch a graph of f z 0 5z 21 1 Before sketching the graph determine where the function has its minimum or maximum value so you can place your first point there 5 4 3 2 1 5 4 3 2 1 Clear All Draw 1 2 3 4 5 2 The zeros of the function are at the values The x intercept s are at the points The y intercept is at the point

1 3 3 Analyze the polynomial function f x 7x7x 7x 21 Answer parts a through h a Determine the end behavior of the graph of the function 1 3 X for large values of x The graph of f behaves like y www b Find the x and y intercepts of the graph of the function The x intercept s is are Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answe only once h

Prove the identity tan x 1 3 x T 3 1 3 tan x 1 3 tan x tan x tan 3 3 tan x 1 3 tan x

Prove the identity sin x sin x 2

Complete the identity csc 0 1

Solve the right triangle shown in the figure A B C a 9 C 69 16 B C O O b C

For the polynomial function f x 6 x 7 x 6 answer the following questions a List each real zero and its multiplicity b Determine whether the graph crosses or touches the x axis at each x intercept c Determine the maximum number of turning points on the graph d Determine the end behavior that is find the power function that the graph of f resembles for large values of x B The smallest zero of f is 6 with multiplicity 2 The largest zero of f is 7 with multiplicity 1 Type an exact answer using radicals as needed Type integers or fractions OC The smallest zero of fis with multiplicity The middle zero of f is Type an exact answer using radicals as needed Type integers or fractions with multiplicity The largest zero of f is D There are no real zeros b Select the correct choice below and if necessary fill in the answer box es to complete your choice Points 0 5 of 1 c The maximum number of turning points on the graph is Type a whole number with multiplicity OA The graph crosses the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed B The graph touches the x axis at Type an exact answer using radicals as needed Type an integer or a simplified fraction Use a comma to separate answers as needed C The graph touches the x axis at 6 and crosses at 7 Type an exact answer using radicals as needed Type integers or simplified fractions Use a comma to separate answers as needed OD The graph neither crosses nor touches the x axis

Complete the sentence below 5 If f x 3x x 2 6x 9 then f x 5 3 If f x 3x x 6x 9 then f x as as x and f x as x and f x as x as x

h x 2x5 Choose the correct graph below OA 5 M B Ay Q OC 1039 O D

Find the point x y on the unit circle that corresponds to the real number t ST

Evaluate if possible the sine cosine and tangent of the real number t If not possible enter IMPOSSIBLE 4 sin t cos t tan t

10 10 5 5 5 Match each graph to the expression that transforms the black solid graph into the blue dashed graph 1 5 5 0 5 0 1 1 7 1 1 1 5 5 5 5 10 10 10 10 x 2 x 2 x 1 2 x 10 10 10 10 5 5 1 5 5 0 0 5 5 10 10 10

3 Zx zy y 17 15 8x 7 y 13 0 5 6

Match the graph of the function z 2 x y to the system of the level curves Which of the following level curves matches the given surface graph OA OB X C SEM SAVAN SVE TERO mack PERGER ID OC

An object dropped from a height of 500 feet has a height h t in feet after t seconds have elapsed such that h t 500 16t Express t as a function of height h and find the time to reach a height of 100 feet t h time to reach height of 100 feet 4

A machinist must produce a bearing that is within 0 02 inches of the correct diameter of 5 5 inches What are the possible diameters that will work Enter solution as an interval or union of intervals help intervals In absolute value notation where a represents the machined diameter this is OA 5 51 0 02 OB x 0 02 5 5 OC 5 5 0 02 OD x 0 02 5 5 OE None of the above

Find the x intercept s and y intercept s of y 2x 5 2 Enter a point as a b including the parentheses If there is more than one correct answer enter a comma separated list of points x intercept s y intercept s help points help points

Find the interval s of numbers that are at least a distance of 8 away from 7 Enter solution as an interval or union of intervals inf 15 U 15 inf A help intervals In absolute value notation where x represents the numbers this is A x 81 7 B x 812 7 C x 7 8 D x 7 8 E None of the above