Calculus Questions

Use Euler s method with the given step size to estimate y 1 4 where y x is the solution of the initial value problem y x xy y 1 0 1 Estimate y 1 4 with a step size Ax 0 2 Answer y 1 4 2 Estimate y 1 4 with a step size Ax 0 1 Answer 11 B

Use the symbols provided below to fill in the phase line for the differential equation x x 3 x 1 To place a symbol on the line Click a symbol button then click a point on the line To remove a symbol from the line Click the delete X button then click the symbol on the line decreasing steady state slope undefined increasing delete X

For each of the following differential equations determine if the proposed function is a solution a Let C and k be constants Given the differential equation f t kf t the function f t Cet is b Given the differential equation 3y the function y 2e 3 is c Given the differential equation f t 1 f t the function f t e t is 2y d Given the differential equation 2 the function y t 3t is a solution

Is each of the following functions a solution to the differential equation y 11y 30y 0 Yes Ma 1 y 3ebm 2 y 6e 5w 3 y There are only 4 attempts allowed for this question 6a 7e 0

For what positive value of k does the function y sin kt satisfy the differential equation y 169y 0 For what negative value of k does the function y cos kt satisfy the differential equation y 169y 0

Find all values of r so that the function y x solves the differential equation x y 6xy 14y 0 into the equation and find values of r that satisfy the resulting equation Hint Plug y T If there are more than one answer use commas to separate the answere

Consider the slope field shown a For the solution that satisfies y 0 0 sketch the solution curve and estimate the following and v 1 y 1 b For the solution that satisfies y 0 1 sketch the solution curve and estimate the following y 0 5 y 1 c For the solution that satisfies y 0 1 sketch the solution curve and estimate the following and and y 1 17 1111 LLLL 1 1 1 1 1 1 111 11 LLL 17 1111

The functions are all solutions of equation y x xy 2y 4x x 0 Find the constant c which produces a solution which also satisfies the initial condition y 5 3

In the spreadsheet pictured an arrow indicates the contents of a cell are copied down its column n the spreadsheet below column A holds different values of h and column B computes 0 h f 0 for the function f x x Writing abs A1 in a spreadsheet will compute the absolute value of the number in cell A1 a What number shows up in cell A6 123456 A 1 A1 10 B abs A1 A1 o f 0 does not exist Select below the best explanation of a way you could suspect that from the spreadsheet The values in column B are not converging to one number The numbers in column B have different signs The values in column A are converging to 0 The values in column A are not converging to 0

Match each differential equation to a function which is a solution FUNCTIONS A y 3x m B y e C y sin x D y t E y 7 exp 8m DIFFERENTIAL EQUATIONS Enter the capital letter that corresponds to one of the functions above eg A 1 On 1 y y 0 2 2m y 3my y 3 y 8y

Each row of the spreadsheet below contains the position of a particle at a given time The times are stored in column B and the corresponding positions in column A Column C should contain the average rate of change of position with respect to time between two consecutive rows To compute these rates of change you will write a formula in cell C1 and copy it to cells C2 C3 C4 and C5 Assume all cells not shown below are blank a What formula should you write in C1 O A2 A1 B2 B1 B2 B1 A2 A1 B 2 B 1 A 2 A 1 B 2 B 1 A 2 A 1 O A 2 A 1 B 2 B 1 A2 A1 B2 B1 A 2 A 1 B 2 B 1 O B2 B1 A2 A1 123456 6 position A004568 time B 0 1 3 4 7 8 average rate of change C diceloved when you press the enter key

Consider the slope field shown a For the solution that satisfies y 0 0 sketch the solution curve and estimate the following and y 1 y 1 b For the solution that satisfies y 0 1 sketch the solution curve and estimate the following E and y 1 y 1 c For the solution that satisfies y 0 1 sketch the solution curve and estimate the following y 1 and y 1 1 2 7 7 7 11 12 12 1 xxx 111 7 11 AVAI 2 4 7 7 7 71 SAVAI p IV X VA 11121 AIZLI W l l l l l l L 7 7 7

In the figure below which of the following is a solution to the differential equation dy dt with initial value y 0 3 4 Note pay attention to axis labels 1 0 8 25 0 5y IV 1 I I 1 I 1 1 I I 1 1 1 0 I 1 I 0 25 II IV

Check by differentiation that y 5 cos 2t 5 sin 2t is a solution to y 4y 0 by finding the terms in the sum So 1 411

Verify that every member of the family of functions y following questions Answer y log x C X 1 Find a solution of the differential equation that satisfies the initial condition y 3 10 is a solution of the differential equation x2y xy 1 Answer the Answer 2 Find a solution of the differential equation that satisfies the initial condition y 10 3

Which of the following functions are solutions of the differential equation y 3y 18y 0 Select all that apply 3x A y x B y x e C y x er D y x e 3r E y x e F y x 6x G y x 0

Let y 16y 0 Find all values of r such that y separated list ker satisfies the differential equation If there is more than one correct answer enter your answers as a comma help numbers

This problem is an example of a differential equation an equation that relates a function to one or more of its derivatives You can solve this problem by doing some educated guessing Educated means remember what we did in the past Suppose f is the function that satisfies for all a in its domain and Then f x f x f 1 1 f x Hint try some of your familiar functions parabolas hyperbolas exponential trig nowor function

Use the given graph of y f x to find the intervals on which f x 0 the intervals on which f x 0 and the values of x for which f x 0 Sketch a possible graph of y f x www OA X On what subinterval s is f x 0 Select the correct choice below and if necessary fill in the answer box to comp your choice Type your answer using interval notation Use a comma to separate answers as needed OB There are no such subintervals

8 Using the second derivative test find the maximum and minimum points for the function f x x 8x 5 What conclusion can be made if a A function changes from a decreasing interval to an increasing interval b lim f x and lim f x 0 x 0 x 0 If the minimum of the function y x 4x 21 occurs at x 2 what are the coordinates of the minimum point

Helium is pumped into a spherical balloon at a constant rate of 3 cubic feet per second How fast is the radius increasing after 2 minutes At what time if any is the radius increasing at a rate of 140 feet per second The radius is increasing at a rate of ft sec Type an integer or a decimal Do not round until the final answer Then round to four decimal places as needed

Find an equation for the parabola that has its vertex at the origin and has its focus at the point 8 75 0

Use the curve sketching procedure to analyze the function a y x x 20x 1 x 1 x Determine the horizontal asymptote x 2x 6 2 x b y a y b y x 3x x 7 x5 3x 8 2x x 3x 4 c f x

Find the vertex focus and directrix for the following parabolas x 2 8 y 2 Vertex Focus

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 271 1 Ost 18 C Sketcn the grapns or N and on the same coordinate system Choose the correct grapn below O C O D A Ay 4 000 0 0 18 O 5 B 4 000 Ay 0 D What is the maximum rate of growth The maximum rate of growth at minutes is 18 Q Q 4 000 0 bacteria per minute 18 Q Ay 4 000 0 0 18 O O 5

A T shirt manufacturer is planning to expand its workforce It estimates that the number of T shirts produced by new workers is given by T x 0 5x 18x 0 x 27 When is the rate of change of T shirt production increa and when is it decreasing What is the point of diminishing returns and the maximum rate of change for T shirt production Graph T and T on the same coordinate system The rate of change of T shirt production is increasing on Type your answer in interval notation

Find the intercepts of the function a f x x 2x 15 x 10 x 5 Determine if the function is even odd or neither b y a f x 1 x x b y x x 7 c f x 3 X X State the domain of the function x 3x 2 X 4 a y b y 4x 7x 13x 100 c f x 2x 10 Determine the vertical asymptote s x 5x 7 x 3x 28 a y b f x x 4x 4x 6 x 2x 5x 6

The marketing research department of a computer company used a large city to test market the firm s new laptop The department found the relationship between price p dollars per unit and the demand x units per week was given approximately by the following equation p 1050 0 14x 2 0 x 75 So weekly revenue can be approximated by the following equation R x xp 1050x 0 14x 0 x 75 Use this information to answer parts A and B B There is no local maximum What is are the local minimum a Select the correct choice below and if necessary fill in the answer box to complete your choice OA The local minimum a is are at x Simplify your answer Use a comma to separate answers as needed

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 27t t Osts 18 Dox to complete your choice A The rate of growth is increasing on 0 9 Type your answer in interval notation Use a comma to separate answer as needed B The rate of growth is never increasing When is the rate of growth decreasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The rate of growth is decreasing on Type your answer in interval notation Use a comma to separate answer as needed OB The rate of growth is never decreasing

A drug that stimulates reproduction is introduced into a colony of bacteria After t minutes the number of bacteria is given approximately by the following equation Use the equation to answer parts A through D N t 1000 27t t 0 t 18 A When is the rate of growth N t increasing Select the correct choice below and if necessary fill in the answer box to complete your choice OA The rate of growth is increasing on Type your answer in interval notation Use a comma to separate answer as needed OB The rate of growth is never increasing

For each function determine i the critical values ii the intervals of increasing or decreasing iii the maximum and minimum points a f x 4x 12x 7 b f x x 9x 24x 10 x 2 x 2x 15 c y 1 Find the intervals of concavity for the function and state the points of inflectio a f x x 2x x 2 x x 4 b f x

K Use the definition of the derivative to evaluate the limit 7 h 343 lim Im 1 0 7 h h Question 24 7 3 343

Use both the washer method and the shell method to find the volume of the solid that is generated when the region in the first quadrant bounded by y x y 25 and x 0 is revolved about the line y 5 O A OB A 5 B 0 x 900 x 5 SCH 0 dy Set up the integral that gives the volume of the solid as a single integral if possible using the shell method Select the correct choice below and fill in any answer boxes within your choice Type exact answers 25 S 2x y 5 y dy 0 SO 0 The volume is dx dx Save

Find the higher order derivative dx 6log x 4 6log x 4 Question 19

OA If b 0 and b 1 then c If b 0 and b 1 then d X OB If b 0 and b 1 thenx logbx Inb OD If b 0 and b 1 then d d dx logbx x in b for x 0 and dx logplx x Inb forx 0 dx d dx OB log bx OA 108x x Inx b dx x 109x x Inx dx OC log x In bx dx dx log bx hic function f x log x How does it differ from the derivative formula for Inx In b for x 0 and In Inb 1 x In b In bi d In b for x 0 and log x for x 0 dx X How does the rule of differentiation for the function f x log x differ from the derivative formula for Inx dx logolx X for x 0 and d dx In b BETER for x 0 1 logxxlnb for x 0

K Evaluate the following integral dx x 11 x 0 dxa Question 21 7

Find the higher order derivative 3x in 3x 3x in 3x 0 Question 20 7

K Use the definition of the derivative to evaluate the limit lim Inx 6 x X 2 6 Type the exact value of the limit lim Inx 6 Question 23 7 3 1

vative y sin 2x In sin 2x Y 0 Use parentheses to clearly denote the argument of each function www 17 7 3 65

K Evaluate the derivative of the following function H x 2x 5 H Question 16 7 3 61

the following integral J5 1 5 1 Inxx dx fax 6 1 In xxx dx Type an exact answer in simplified form

K Find the derivative of the following function yain x 8 dy dx Type an exact answer using it as needed Question 12 7

Evaluate the derivative y 3x y Type an exact answer Question 15 7

of the function and evaluate the derivative at the given value of a f x sin x a Find the derivative of the function sin x x dx Use parentheses to clearly denote the argument of each function

V Find the derivative of the following function y 22 sinx dx 22 s n x Question 5 7 3

Evaluate the following integral 5 dx 4x 54 dx 4x 5 In 5

y x 45 dy dx Type an exact answer using radicals as needed N

11 214 the following integral 1Aa dp dp Type an exact answer 7 3

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K Solve the following equation Simplify your answer Use a comma to separate answers as needed