Differentiation Questions and Answers

If r t cos 5t i sin 5t j 10tk compute r t and fr t dt with C a constant vector i i j j

In Seattle on September 30 the temperature in F t hours after midnight was given by the function Temp t 60 12 sin t 11 What was the average temperature over the period from 8 A M until 10 P M OA 76 39 F OB 60 F O c 56 05 F OD 76 39 F OE 65 48 F

1 Use the eigenvalue to find the general Method and eigen vector solution of the following finst onden system where x Ax Show all your work X 2X 5X2 X 4x X

7 Solve the following initial value phroblem the Laplace Transform Alternate Use Show all work do not X x 12X 12x 0 10 Or methods

You may use your calculator on the following three questions 11 Consider a curve given implicitly by the equation 1 x y xy 85 0 A Calculated at a general point x y B Write the equation of the tangent line to the curve at the point 3 1 C At 3 1 y x is defined implicitly as a function of x Let g x be the inverse function of y x Compute g 1

16 Suppose that f has a continuous second derivative for all x and that f 0 1 f 0 2 and f 0 0 A Does f have an inflection point at x 0 Explain your answer B Let g x 3x 2 f x x 2x 5 f x The point 0 5 is on the graph of g Write the equation of the tangent line to g at this point C Use your tangent line to approximate g 0 3 D Find q 0

You may NOT use your calculator on the following three questions 14 S 4 3 2 H Let f x M 2 3 1 g x Let g be a function whose domain is the closed interval 0 8 The graph of g is shown above The graph is made up of straight line segments and a semicircle 3 4 5 6 7 8 2x 1 g t dt A Compute f C Find f 3 B What is the domain of f D Find the x coordinate of the absolute maximum of f on its domain

velocity by time it hits the ground By measuring the impact you determine that this terminal velocity is 49 m sec A Write the equation representing the velocity 1 of the object at time seconds given the initial velocity vo and the fact that acceleration due to gravity is 9 8 m sec2 Here assume you re modeling the falling body with the differential equation g kv and use the resulting formula for v 1 found in the Tutorial Of course you can derive it if you d like dv dt B Determine the value of k the continuous percentage growth rate from the velocity equation by utilizing the information given concerning the terminal velocity

3 Biologists stocked a lake with 800 fish and estimated the carrying capacity of the lake to be 20000 In the first year the number of fish doubled Use a logistic model to model the fish population and answer the questions below A Find an expression for the size of the fish population P P t at time in years You may approximate the constant to six decimal places B How long will it take for the fish population to reach 10000

K if f x 4 1 If f x compute f 2 and f 2 9 f 2 Simplify your answer 1 16 f 2 Simplify your answer

Compute the following 1 f 2 and f 2 when f t 2 t f 2 Type an integer or a simplified fraction

Suppose the revenue from producing and selling x units of a product is given by R x 6x 02x dollars a Find the marginal revenue at a production level of 20 b Find the production levels where the revenue is 400 a The marginal revenue at a production level of 20 is dollars per unit

Find the first and second derivatives y x 11 1 2 x 11 y y 0

8 For the function A whose graph is shown state the following a lim A x b lim A x x x 3 c lim A x x 2 e The equations of the vertical asymptotes 3 d lim A x x 1 YA 0 I 2 5 18

The point P 2 1 lies on the curve y 1 1 x a If Q is the point x 1 1 x find the slope of the secant line PQ correct to six decimal places for the following values of r ii 1 9 vi 2 1 b Using the results of part a guess the value of the slope of the tangent line to the curve at P 2 1 i 1 5 v 2 5 iii 1 99 vii 2 01 iv 1 999 viii 2 001 c Using the slope from part b find an equation of the tangent line to the curve at P 2 1

pi 1 Find the derivative of f x 3 10 at x 3 10 Type an integer or a simplified fraction

If f x 1 10 X compute f 2 and f 2 f 2 Simplify your answer

Differentiate d dx y 3x 8 5 3x 85

Differentiate d dx 6 f x x x x 4 x x x

Find an equation of the tangent line to the graph of y f x at the given x 1 49 f x x x What is the equation of the tangent line Type your answer in point slope form

Use the power rule to find the derivative of f x x6

Solve by the quadratic formula and simplify 3x x 2 0 X Use a comma to separate answers as needed

Suppose A is a 3 x 3 matrix and B 2A Then i B 8 A ii B 4 A iii B 2 A iv B A V None of the options

If the graph of a x is horizontally compressed by one fourth to obtain the graph of b x fill in the table for b x y a x X X Y Y 4 2 1 16 12 10 After you pick your answer press GO y b x 16 12 0 8 10 4 0 8 0 8 8 8

2s 7 7s 12 a Find the partial fraction decomposition of F s 1 point Consider the function F s 2s 7 5 7s 12 b Find the inverse Laplace transform of F s f t C F s help t

Consider the curve given by x2 sin xy 3y2 C where Cis a constant The point 1 1 lies on this curve Use the tangent line approximation to approximate the y coordinate when x 1 01 OA 0 996 OB 1 O C 1 004 O D Cannot be determined OE 1 388 C

The position feet traveled of a car is given by the equation s t t 1 Find the time when the car is going the same speed as its average speed over the interval 0 to 10 seconds A t 2 5 B t 5 OC Never D t 0 OE t 10

function f t defined on an interval 0 t L is given Find the Fourier cosine and sine series off and sketch the graphs of the two extensions of f to which these two series converg f t sin 0 t 4

M Which of the following represents the graph of f lx like uns OA None of these OB C OD V A

Find the indicated root of the given equation to at least four decimal places by using Newton s method Compare with the value of the root found using a graphing calcu 5x4 6x 5x 2x 3 0 between 1 and 0 The solution obtained by using Newton s method is approximately Round the final answer to four decimal places as needed Round all intermediate values to seven decimal places as needed

When a circular plate of metal is heated in an oven its radius increases at a rate of 0 03 cm min At what rate is the plate s area increasing when the radius is 41 c cm min The rate of change of the area is Type an exact answer in terms of

Which of the following could be the units for if y is the surface area of a tumor and x is the radius of the tumor O A Millimeters per centimeter OB Meters per radian O C Meters per second OD Gallons per hour E Square millimeters per millimeter

Use the Runge Kutta implementation in Matlab ode45 to find a numerical solution y 0 6 to the differential equation dy t y t 3 sin 4t y t t y 0 0 dt Give your answer correct to 3 decimal places Number Note the Matlab in the lecture notes produces a plot of the solution to get a numerical answer just leave out the plot command and leave out the semicolon at the end of the ode command then the numerical values will be output to the screen The last one is the result of the last step

1 For each of the following find lim f x x a I II III lim f x x a lim f x x a A f x 2 x 3x 2 c f x 1 2 x 4 sinx if x B f x tanx if x cosx if x X 3 x 3 2 s n 2 for a 2 T TC TT if x if x T a TO a

The sides of the triangle shown increase in such a way that dz dx dy 1 and 3 At the instant when x 12 and y 5 what is the value dt dt dt of y dx dt X O A 39 41 OB 27 O C 13 OD 13 OE 13 E 17 27 13 41 N

Find the length of OR R 6 1 9 OR C

hours The cost for each firefighter is 20 for transportation plus 25 per hour for salary meals and other overhead A Let C be the cost of the fire including the cost of the firefighters and the lost timber C will depend on both h how many hours the fire burns and x the number of firefighters sent to fight it Give a formula for C in terms of x and h I

is a function write yes or no neither 3 Domain Range Function

Use this graph of f to find the solutions to f x 1 104 10 8 6 4 2 8 6 4 2 0 2 4 6 8 G 2 4 6 8 Ag 10

Use tangent line approximation to estimate 2390 to seven decimal places recognizing that 7 2401 Show all your work

5 Use tangent line approximation to derive an estimate for 1 x when x is near 0 and n is any real number

3 Find the point on the graph of y x2 1 that s closest to the point 8 1 5 Hint Remember the distance formula The distance between two points in a plane is d x x 12 2 2

6 Use the graph of y g x below to estimate point s of inflection for the function g x as well as the intervals on which g x is concave up and concave down Explain your reasoning 1 6

3 Consider the function f x 2x 3x 6 x 2 A Find f x Show your work B Graph f x on your calculator and use this graph to answer the following questions On what interval s is f x concave up and how did you use the graph of f x to estimate On what interval s is f x concave down and how did you use the graph of f x to estima this Note You don t need to include a sketch of y f x in your answer

5 Below is a graph of the curve y 4x cosx 10 10T SIX 10 Your friend claims that the slope of the tangent line to the curve at x 4 is the negative of th slope of the tangent line to the curve at x 4 Prove mathematically whether your friend is right or wrong Is the slope of the tangent line to the curve at x a the negative of the slope of the tangent to the curve at x a for any other values of a Explain

1 Find the slope of the tangent line to the curve y x3 2x 1 at the point 1 0 Find the equation of that tangent line On your calculator graph both the curve and the tangent line on the same set of axes with viewing window x 2 3 y 2 2 Show all work except for anything concerning the graphs you made on your calculator they were for your observation only I 2 Find the slope of the tangent line to the curve y at the point 1 1 Find the equation of that tangent line On your calculator graph both the curve and the tangent line on the same set of axes with 11 523 Show all work oxcont for anythin ho aranhe

37 If f x sec x then lim X A 0 B sec C sec tan 5 D nonexistent 15 00 f x f 5 T X 3 is

C Write an equation for C then state the slope of the graph and determine its interpretation in the context of the problem Total Cost for Electrician Services 330 900 850 Boo 730 00 6gn ban 49 300 450 400 350 300 230 ann 200 54 o 0 15 Number of Hours Working The slope of the function is which represents the number of hours the electrician spends at the house the set fee for the house call the hourly rate charged H MET CERE S O O d U

Given z 2 xy x uv w y u ve z du then find 20 043 when u 1 v 3 w 0

If f x sin 7 5x find f 7 O A 0 657 OB 0 754 O C 0 OD 0 657 OE 3 770