Calculus Questions

On what domain is the function f x x 3x 4 1 x 1 continuous

f x Get a similar question x 6x 5 x2 3x 4 f 5 lim f x x115 f x f x is continuous lim f x x 5 Add Work 0 You can

2 H 10 9 8 6 5 4 3 a Give the intervals where the function is continuous Answers are expected in interval notation e g 00 2 U 2 00 Use U between intervals and use oc oo for negative and positive infinity and are differentiated Unless explicitly shown in the image ass the left and right endpoints are finite and closed

Use the figure below which gives a graph of the function f x to give values for the indicated limits Enter NA if a limit does not exist 2 8 a lim f x b lim f x c lim f x d lim f x 6 4 2 6 8 10 1 O 2 3 4 O 5

Calculate the gradient of h x y z x 2 Vh 3x 4yz3 x 3z3 3yz2 Answers Attempt 8 of 8

Find the exact length of the curve of r t sint cost In cost for 0 t 7 7 points

C A e AL IlaInst 3 Above are 6 surfaces A F Match them with the following equations HINT Think about what the traces would look like 6 points a y z 0 b 2x 3y z 5 4x 2 z 1 D x 2 2 9 F d x z 1 f y x

BONUS 3 Find a vector in the direction of 4 8 8 with a length of 10 Must show work to receive credit

Find the curvature of x t y 2t3 at the point 1 2 Round your answer to 4 decimal places 7 points

4 Find the exact value of t such that r t 5 if r t 4et 5 6e t 7 points

5 Given r t 10t 1 t for o t o find a vector orthogonal to r and r 7 points

For each statement choose TRUE or FALSE No work needed 2 points If k t 0 for all t then the curve is a straight line TRUE FALSE If u xv 0 then u 0 or v 0 TRUE FALSE

Let f x y x y and c t t 2t a Calculate Vf c t 3x y 21 x 61 b Use the Chain Rule for Paths to evaluate f c t at t 1 f c 1 18 Answers Attempt 5 of 5 Answer

a Find the critical point of f x y 2x y z 4y x y 6x b Find the discriminant D x y for the function D x y 0 what is the value of the discriminant at the critical point D 48 0 14 0 8 14 0 What does this tell us about the critical point The discriminant doesn t tell us anything c Sketch contours near the critical point to determine or confirm whether it is a local maximum a local minimum a saddle point or none of th The critical point is a saddle point

a Find the critical point of f x y 2x y 2 4y x y b Find the discriminant D x y for the function D x y what is the value of the discriminant at the critical point D What does this tell us about the critical point The discriminant doesn t tell us anything c Sketch contours near the critical point to determine or confirm whether it is a local maximum a local minimum a saddle point or none The critical point is a saddle point

The figure below shows the level curves of a function f x y around a maximum or minimum M The 2 and y axes are along the bottom and left edge of the figure respectively One of the points P and Q has coordinates 1 1 and the other has coordinates 22 2 Suppose b 0 and e 0 Consider the two linear approximations to f given by f x y a b z z1 c y v and a What is the relationship between the values of a and k Give your answer as an equation b What are the coordinates of P yl c Is M a maximum or a minimum minimum d What can you say about the sign of the constants m and n THE 0 and f x y k m z z2 n y

Find the critical points for the function and classify each as a local maximum local minimum saddle point or none of these critical points give your points as a comma separated list of x y coordinates classifications 4 give your answers in a comma separated list specifying maximum minim f x y 32xy x y

For f x 3x 4 which of the following domain restrictions will allow f x to be invertible Select Which of the following is f x Select A B C x 4 3 x 4 3 x 4 3 D 1 x 4

Here is a contour plot of the function f a 4 3y Click the image to enlarge it By looking at the contour plot characterize the two critical points of the function You should be able to do this analysis without computing derivatives but you may want to compute them to corroborate intuition The critical point 1 1 is a local The second critical point is at the point choose one from the list and it is a saddle point 0

Given the graphs of f x and g x find g f 4 5 0 5 f x g x

Find the maximum rate of change of f 2 2 3 2 at the point 4 4 1 and the direction in which it occurs Maximum rate of change Direction unit vector in which it occurs

Calculate the gradient of h x y z x 2 23 Vh A B 5 3 Z 3x x 122 1 2 3x2 23

Evaluate the following Find the arc length of the graph of y on the interval 1 2 To enter your answer click on the three vertical dots on the right of the task bar then click on the symbol This will bring up the equation editor Use the symbols to write the mathematical sentence Work will be graded separately Your Answer 11 S T

Draw a picture and set up the integral using the method specified DO NOT INTEGRATE Draw a picture and set up the integral to find the volume of the solid generated by revolving the region bounded by the graph of x y 16 x 0 about the x axis Shell Method To enter your answer click on the three vertical dots on the right of the task bar then click on the symbol This will bring up the equation editor Use the symbols to write the mathematical sentence Drawing and work will be graded separately Your Answer

Click here to begin Submit Answer 20 Points lim X 2 Evaluate the limit if it exists If an answer does not exist enter DNE 8x 4 x 2 Submit Answer DETAILS 20 Points SCALCET9 2 3 015 DETAILS SCALCET9 2 3 029 Evaluate the limit if it exists If an answer does not exist enter DNE

Evaluate the following Find the volume of the solid formed by revolving the region bounded by the graphs of y 1 and y 2x about the line y 1 To enter your answer click on the three vertical dots on the right of the task bar then click on the symbol This will bring up the equation editor Use the symbols to write the mathematical sentence Work will be graded separately Your Answer 74T 15

Adderall is a proprietary combination of amphetamine salts that is used to treat ADHD Attention Deficit Hyperactivity Disorder The patient takes one pill before 8am and the drug has completely entered the bloodstream by 8am At 8am the blood concentration of the drug is 33 8 ng ml Adderall has first order elimination kinetics 8 with 7 7 of the drug being removed from the blood each hour I a The concentration of the drug in the patient s blood t hours after 8am is measured to be Ct Write a recursive relation of C 1 in terms of Ct Assume that the patience takes no other Adderall pills What happens in the long run to the amount of Adderall in the system b Solve your recurrence equation to derive an explicit formula for Ct as a function of t When does the concentration of drug first drop below 0 1 ng ml

2 7 pts Equation of a Line Write an equation of the line passing through the points 1 2 and 4 10

1 Difference Quotient Simplify the difference quotient A 6 pts f x 3x 2 B 7 pts f x dx cx b f x h f x h for each of the following functions

3 2 4 Ex 1 pp63 64 YA 6 5 2 3 4 y f x 5 6 X 1 2 3 4 5 6 7 8 9 Domain Range Function f 1 f 2 f 3 lim f x lim f x lim f x X 2 10 lim f x X 3

Problem Set 1 In the figure to the right AB IDE and BC EF Prove that m ABC mZDEF Evidence Claims 1 ABIIDE BC IEF 2 Draw EZ label z 3 m ABC m E2C 4 m EZC M DEF m 5 mx ABC M4 DEF 1 AB II CO 2 Draw FE label w vo 1 Given 2 Construction 3 4 2 In the figure to the right AB CD Prove that mZAEC a c 3 a w Co 4 w m AEC 5 a c m AEC 5 1 Given 2 Construction 3 linear 4 Angle addition 5 substitution a B A D wo Ivo E C Co

7 Instantaneous velocity Consider the position function s t 1612 128t Exercise 13 Complete the following table with the appropriate average velocities Then make a conjecture about the value of the instantaneous velocity at t 1 1 2 1 1 5 1 1 1 1 1 01 1 1 001 Time interval Average velocity

Instantaneous velocity Consider the position function t 1612 100t Complete the following table with the appropriate average velocities Then make a conjecture about th value of the instantaneous velocity at t 3 Time interval Average velocity 2 3 2 9 3 2 99 3 2 999 3 2 9999 3

11 Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph

x Cos y Use derivatives and exact partial derivatives at 1 56 2 1 Use the finite difference formulas to estimate af and off at 1 56 2 1 for three different x dy values of the mesh size differentiation rules to find the exact partial af f x and evaluate those o h 0 01 o h 0 001 o h 0 0001 dy 9

17 E n 1 1 2 3

A city has streets that run north and south and avenues that run east and west all equally spaced Streets and avenues are numbered sequentially as shown in the figure The walking distance between points N W E 1st St 3 blocks 2nd St 4 blocks 5th St 7th Ave 6th Ave 5th Ave 4th Ave 3rd Ave 2nd Ave 1st Ave a Find the straight line distance in blocks between A and B blocks b Find the walking distance and the straight line distance between the corner of 3rd St and 6th Ave and the corner of 15th St and 1th Ave walking distance blocks straight line distance blocks c What must be true about the points P and Q if the walking distance between P and Q equals the straight line distance between P and Q O Points P and Q must be on the same street and the same avenue O Points P and Q must be on different streets and different avenues OPoints P and Q must either be on the same street or the same avenue O Points P and Q must either be on different streets or different avenues

Suppose f x y tan x y and u is the unit vector in the direction of 3 2 Then a Vf x y b Vf 0 5 9 c fu 0 5 9 Du f 0 5 9

Determine the global extreme values of the function f x y 4x 4x y 2y x y 20 x y fmin

Let f x y xey and P 5 25 a Calculate Vfp l b Find the rate of change of f in the direction Vfp c Find the rate of change of f in the direction of a vector making an angle of 45 with Vfp Answers b c CAR P AMD

Calculate the directional derivative of f x y z y in the direction of vi 2j at the point P 2 2 Remember to normalize the direction vector Duf 2 2 12

Find the critical point of the function f x y x y 4xy 42x C Use the Second Derivative Test to determine whether the point is O A a local maximum OB a local minimum OC test fails O D a saddle point

If f x y z 2zy then the gradient at the point 3 2 4 is Vf 3 2 4 B

The function f has continuous second derivatives and a critical point at 10 1 Suppose faz 10 1 2 f 10 1 1 f 10 1 4 Then the point 10 1 O A is a saddle point OB cannot be determined OC is a local maximum OD is a local minimum OE None of the above

Determine the global extreme values of the function on the given set without using calculus f x y 2e 2 8y z 8y 1 fmin

Calculate the directional derivative of f z y sin y in the direction of v 1 23 at the point P 0 5 0 5 Remember to normalize the direction vector D f 0 5 0 5

Consider the function f x y cos 9y Find and classify all critical points of the function If there are more blanks than critical points leave the remaining entries blank f fay 4 1

Find the equation of the circle shown in the figure LAHA 3 2 1 y 3 1 1 2 X

Make a table of values and sketch the graph of the equation Find the x and y intercepts and a x 3y 6 Make a table of values and sketch the graph of the equation y 10 10 5 y intercept x y 5 10 5 5 10 y 10 5 5 10 5 Test for symmetry Select all that annly 5 10 10 X Find the x and y intercepts If an answer does exist enter DNE X intercept x y X O 11 10 5

Find the critical points of the function f x y x y 144xy Answer 8 Starting with the point with the smallest z value use the Second Derivative Test to determine whether each critical point is P O A a saddle point O B test fails OC a local minimum O D a local maximum P O A a local maximum OB a saddle point O C a local minimum OD test fails P3 O A a local minimum O B a local maximum O C test fails O D a saddle point