Differentiation Questions and Answers

find 14 Use Newton s method with the specified initial approximation to X39 the third approximation to the root of the given equation Give your answer to six decimal places f x x x 1 x 1 5

Consider the following function f x 2x a 0 n 3 0 x 0 2 Exercise a Approximate f by a Taylor polynomial with degree n at the number a Step 1 The Taylor polynomial with degree n 3 is T3 x f a f a x a The function f x 2x Step 2 f x f x f x 4x 16x 4 With a 0 f 0 has derivatives 64x 48x f 0 Submit Skip you cannot come back Step 1 If Ir 4 x M then we know that M IR x 4r a Ix 16r 4 e x 64 48 and Je x Exercise b Use Taylor s Inequality to estimate the accuracy of the approximation fx T x when x lies in the given interval f 0 and f 0 0 4

Find the exact length of the polar curve described by r 9e 0 on the interval 0 4 6 T 1 tan Y I syntax incomplete

Question 12 Match each graph with its equation V 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 4 3 2 1 2 3 4 2 4 2 3 4 4 3 2 N H 22 3 4 4 3 A 3 4 4 2 a 12 3 4 d a r 1 2 cos 0 b r 3 sin 20 c r 3 cos 20 d r 1 2 sin 0 e r 3 cos 30

Let f u logo Determine d du f u Hint d du 7 3 f u

Convert the point with Cartesian coordinates 4 4 3 to polar coordinates Enter exact values for r and 0 with r 0 and 0 0 2 in radians r 0 Question Help Video Morrago instructor

Find the present value of a continuous income stream of 40 000 per year for 35 years if money can earn a 0 8 annual interest compounded continuously b 2 5 annual interest compounded continuously c 4 5 annual interest compounded continuously

14 Let f x y z e y z xyz and g t f at 1 bt 2 ct Suppose there is 0 such that g is decreasing on 0 e Then a b c can be A 3 4 12 B 3 4 12 C 0 3 1 D 0 3 1

In Exercises 39 42 compute the Taylor polynomial indicated and use the Error Bound to find the maximum possible size of the error Verify your result with a calculator 39 f x cos x a 0 cos 0 25 T5 0 25 40 f x x 2 a 1 f 1 2 T4 1 2

19 Show that the nth Maclaurin polynomial for f x e is X X Tn 2 1 1 2 xn nl

b 6 4 4 2 2 4 6 2 4 6 X Yes the vertical line test shows that the graph represents y as a function of x Yes the horizontal line test shows that the graph represents y as a function of x No the vertical line test shows that the graph does not represent y as a function of x No the horizontal line test shows that the graph does not represent y as a function of x There is no way to determine this using the graph

Functions Course Packet on cost revenue and profit functions A manufacturer of soccer balls has a monthly fixed cost of 100 and a processing cost of 5 for each soccer ball produced If x represents the number of soccer balls produced sold determine the total cost function C x Total Cost function C x If each soccer ball sells for 35 determine the revenue function R x Revenue function R x Determine the total profit function P x Total profit function P x Calculate the total monthly profit when 200 soccer balls are produced and sold each month Profit dollars

Unctions domain and range Determine the domain of each of the following functions Enter your answer using interval notation a Domain of 8x 5 b Domain of 8x 5 c Domain of 5 8x d Domain of 5 8x 50 80 8 8 588 5 Your answer is incorrect

which statement accurately descr 1 x 1 1 8 x 1 asymptotes of the functions below A The horizontal asymptote for f x is y 0 The horizontal asymptote for g x is y 1 and y 1 B The horizontal asymptote for f x is y 1 The horizontal asymptote for g x is y 0 C The horizontal asymptotes for both functions is y 1 D The horizontal asymptotes for both functions is y 0 27 Which equation could represent the function graphed below A x B f x C x D x x 12x 32 x 4x x 4x 32 x 4x x 4x 32 2 x 4x x 12x 32 x 4x

6 If h r is a differentiable function with h 1 2 and h 1 3 evaluate f 1 given that T h T 1 ln x f x

Roller Coaster Speed mph Height feet Length feet word Batman the Escape 55 90 2300 a Convert the data set into a matrix b What are the dimensions of the matrix Great White 50 108 2562 Mr Freeze 70 218 1300

B B C D If the distance from D to D is 10 and the distance from A to D is 2 what is the scale factor 1 5 1 6 5

is trying to solve the following If 24 people have the flu out of 360 people how many would have the flu out of 900 Choose EVERY proportion that David could use to solve this problem 360 360 24 300 900 350 900

Use the general solution y of the differential equation to find a particular solution that satisfies the indicated initial condition Graph the particular solutions for x 0 y 6 Cex y 6 y A y 0 2 Ay A Find and graph a particular solution for the initial condition y 0 2 Select the correct choice below and fill in the answer box to complete your choice Type an equation O A 8 0 0 X B y 0 6 8 C y 0 10 OB Ay 10 0 0 X 10 Q O C Ay 10 0 0 X 10 Q O D Ay 10 0 0 X 10

Questi Darlena has started taking photos at amateur dog racing events later offering the photos for sale to the dog owners by email The prices she has charged per photo at each of her first three events and the corresponding number of photos sold and total revenue raised appear in the table below Number of Photos Sold Price per Photo 36 33 21 Revenue 200 Treating revenue as a function of the number of photos sold a graph of the three data points is also shown If she uses quadratic regression to fit a curve to the data what number of photos sold and what price per photo will maximize her revenue 160 120 8 40 5 9 0 Revenue 144 165 189 Enable Zoom Pan 10

After you graduate you get a job as a medical account manager making 63 000 per year Being the wise planner you are you want to begin to save for retirement Your employer gives you the option of investing 8 of your monthly pre tax income into a 401k The account earns monthly compound interest at a rate of 7 25 Determine your monthly investment How much money will you have saved in 40 years when you hope to retire

Find a b a 1 1 b 3 5 2 O 4 5 1 5 5 3 5 2 1 5

Find the trigonometric function value for the angle shown sin e 10 3 2 4 O sin 8 sin 8 sin 8 sin 8 10 5 5 10 HHHHHH 34 2 4 4 34 34 2 34 17 2 2 3 8 5 10 X

Determine two pairs of polar coordinates for the point with 0 0 360 5 5 O 5 2 315 5 2 135 O 5 2 225 5 2 45 O 5 2 45 5 2 225 5 2 135 52 315

d the sum of the following series 9 27 243 the toolbar press ALT F10 PC or ALT FN F1C

Use Matrices A B C and D to answer questions 10 15 when possible When not possible explain using mathematical language 1 228 10 What is C22 11 D 12 C 13 A 14 AB 15 B 2C B 3 7 4 9 12 2 2 0 5 7 C 1 3 3 D 2 2 1 1 01 4 1

Do we want them to do equations like these Solve each equation 1 X 6 6 2 4 9 9 COMMON CORE MATHEMATICS CURRICULUM 9 3 7 0 3 10 5 3Z 3 Y 8 12 12 36 1 4x 10 410 4101 Lesson 7 M5 PRECALCULUS AND ADVANCED TOPICS 2 44 8 40 28 4 2X 16 18 20 4 6 3C 8 2B 4 8 9 0 2 14 7 16 24 33 10 28 2 4 18

COMMON CORE MATHEMATICS CURRICULUM Do we want them to do equations like these Solve each equation 1 X 6 6 2 4 9 9 3 0 3 Y 8 10 12 5 3Z 32 3 2 21 1 4x 1 36 4X 9 10 2 44 8 40 28 Lesson 7 M5 PRECALCULUS AND ADVANCED TOPICS 4 2X 6 3C 1 16 18 20 4 8 2B 4 8 9 0 2 14 7 16 24 33 4

This problem has two parts Part 1 The line y 3x 1 blue dashed 3 2 intersects the curve y x x solid when x 1 What is the value of c c x c red a For what other x values does the line intersect the

Let f w f w 5w W3 Determine the derivative of f 4w 3 Hint Determine the interval s on which f is differentiable Report the solution using interval notation f is differentiable on

Consider the function f x x 4x 4x 1 Find the critical point of f x 2 Find the global maximum 3 Find the global minimum

5 a Convert the expression below into an expression involving log r log y and log z i e write as a sum and or difference of logarithms Simplify

Question 36 The limit below represents f c for some function f and real number c 23 2x 3 x 1 lim 2 1 Determine the function f f x Determine the value of c C Submit All Parts

h x Which of the following is a correct antiderivative Select ALL that apply NOTE in this kind of problem multiple answers or select ALL the grading feels strict You will be awarded a fraction of a point for each correct answer you choose If you select incorrect answers fractional points will also be deducted A student who chooses both correct and incorrect answers can end up with a score of zero Remember your help resources and submit your quiz answers after you discuss problems thoroughly H x Cx x x for any constant value C OH x x x x OH x x x x 1 OH x x x x C for any constant value C r 6r 2

Write the equation of the conic section shown below y 10 9 8 7 6 5 4 V 3 2 1 1 10 9 8 7 6 165 4 3 2 1 2 3 4 5 6 1 2 3 4 I L 5 6 8 9 8 9 10 directrix X

2 Explain why the derivative of g x rule Find the correct g x x 1 dt is NOT g x cos x using FTC1 and the chair di 1 1 1 x tan x

3 A Future Farmers of America club wants to fence a rectangular area using only the 600 feet of fencing that they already own They build the fence so that a barn makes up one side of the area so that side doesn t need fencing Find the dimensions of the fence that they should build so that the maximum possible area is enclosed

p and q are complex numbers such that p 00 9 2 4 5 00 On what interval must O O 9 10 20 fall on 2 10 0 2V 10 00 O 9 q 4 5and p q 9 2 9 2 4 5 00 1 2 3 4 5 6 7 8 9

Use the power series 8 8 1 x 1 x 1 n 0 to determine a power series for the function centered at 0 18 d 9 21 x 1 3 dx x 1 f x x f x 1 n xn n 2 n n 0 X Determine the interval of convergence Enter your answer using interval notation

Criteria for success Expression of the function is simplest form Expression is used to calculate the average et d t 50t feet continue to be the distance function with espect to time in seconds of the car Compute the average elocity of d over the following smaller and smaller intervals 1 1 h where h 1 0 1 0 01 and 0 001 Write an expression for the average velocity from t 1 to t 1 h velocity Answer is rounded to hundredth with correct units

6 Find the derivatives of the following functions a 1 x e ln x b g x sin 10r c h x 3 tan x sin x 5

equation x y 3xy a Find dy dx using implicit differentiation b Determine whether the tangent line at the point x y 3 2 3 2 is horizontal CIRCLE ONE HORIZONTAL NOT HORIZON TAL

A parabola is shown with the focus open circle and a point on the parabola closed circle marked 10 9 8 7 6 5 4 321 The focus F is at 1 2 10 9 8 7 6 5 4 3 217 I 2 45s7WNY 2 3 5 6 O 8 9 10 The directrix has equation y 4 5 6 7 8 9 IC 2 100 o The vertex V is at 1 1 The line of symmetry has equation 1 symmetry The equation of the parabola is x 1 12 y 1 Enter the equation of the directrix Enter the equation of the line of Enter the equation of the parabola

Sketch the graph of 7 6 5 4 3 2 1 2 the cursor until the graph is correct 7 6 S 4 43 2 1 Graph by selecting a type of graph then select the center then movi 3 4 5 6 7 123 6 6

A parabola has focus 6 5 and directrix y 2 Find its x and y intercepts If these do not exist enter DNE The x intercept s are intercept separate the coordinates with a comma The y intercept s are intercept separate the coordinates with a comma Enter the coordinates if there is more than one x Enter the coordinates if there is more than one y

9 9 Gravel is being dumped at a rate of 30 ft3 min It forms a conical pile whose base diameter and height are always equal so 2r h where r radius of the base and h height How fast is the height increasing when the height of the pile is 10 ft The volume of a cone is V r h

Problems 1 Say that the height s of a ball thrown into the air is given by s t 16t 20t 10 a Sketch a graph of s t on the time interval 0 t 2 b Find the average velocity of the ball on the interval 0 t 1 c Find the instantaneous velocity of the ball at t 1 Is the ball going up or going down at that time d Find the equation of the tangent line to the graph of s t at t 1 e On your graph in part a sketch the secant line on the interval 0 t 1 and the tangent line at t 1

3 A farmer wants to fence a rectangular area as inexpensively as possible She builds the fence so that a warehouse makes up one side of the area so that it doesn t need fencing Assume that fencing materials cost 10 dollars per linear foot and that she has 2000 dollars for the project Find the dimensions of the fence that she should build so that the maximum possible area is enclosed

The equilibrium solutions of the differential equation P are always horizontal lines O False O True

Short Answer 1 Let f x be a function State the definition of the derivative of the function at a point o