Calculus Questions

a Change the point 2 3 3 5 which is in rectangular coordinates to cylindrical coordinates Let r 0 and 0 0 2TT b Describe in words the surface whose equation is z 49

Compute the given integral 5x 1 5x 2x 3 dx 40

Find 6 csc 2 da CS fo 6 csc 22 d dx

Find the volume of the solid generated by revolving the shaded region about the y axis The volume of the solid generated by revolving the shaded region about the y axis is 1 0 5 x 2 tan 2 5

Consider the indefinite integral In z dz In 2 Z This can be transformed into a basic integral by letting u du and dz Performing the substitution yields the integral du

Evaluate the following indefinite integral S 3 3t 3 dt

of 11 cos 6x dx Find S 11 cos 6x dx 11 sin 6x 6 a similar question You can Hint Did you include the constant of integration Question Help Written Example Add Work X

Find s S cos 9x dx cos 9x dx sin 9x 9 Hint Did you include the constant of integration X

5 sin 9x 1 dx a This can be transformed into a basic integral by letting Consider the indefinite integral u du and b Performing the substitution yields the integral c Once we integrate and substitute the final answer in terms of x is Question Heln Writton Eva

What is the solution set to log x 7 log3x O 0 3 O 7 0 0 3 3 00 1012 FOTO RIGH

The slope of the tangent line to a curve is given by f x 4x 7x 8 If the point 0 7 is on the curve find an equation of the curve f x

F x y z cos z i 3ysin 3z j 6x zk a Find div F b Find curl F

Solve the initial value problem dy 2e sin e 25 y in 5 0 5 The solution is VE

3 2 1 1 Y 2 1 0 2 graph of f x 3 4 5 6 7 8 9 f 0 f x has a local a Find f x dx dr b Find c Find d Use the graph of f x shown above to find the critical points of f x and to classify each as the location of a local maximum or local minimum f r has a local at the critical point z at the critical point x f x dx F x dx e Suppose f 3 5 Use the Fundamental Theorem of Calculus and the graph of f x shown above to find each of the following f 6

Functions f x and g x are composed to form h x x 2 If f x x 2 and g x x a what is the value of a 0

For the complex number z 150 210 O 240 330 21 2 T 2 i what is arg z

20 points Define g x y x x y 1 Compute the difference between the maxi and minimum of g over the region bounded by the y axis and the circle x y 4

6 5 points The image below depicts which of the following approximations of 10 f x dx A A right hand sum with n 2 and Ar 6 B A left hand sum with n 6 and Ar 2 C A right hand sum with n 6 and Az 2 D A left hand sum with n 2 and Ax 6

7 5 points Which of the following is the right hand sum with n 2 rectangles for f x sin z on interval 2 x 4 A 2 sin 2 sin 0 sin 2 sin 2 B 2 sin 0 sin 2 sin 4 2 sin C 3 sin 2 sin 1 3 sin in 1 D 3 sin 1 sin 4 in 4 sin 4 Esin 1 sin 4

4 5 points Which of the following is a true statement ab Y 6 4 2 f t LA 1 2 3 6 5 4 3 2 2 5 A Ax 8 5 B Ax 4 C Ax 8 D Ax 34 E Ax 32 12 X A on the interval x 20 the function f x has a global minimum at x 3 and a global maximum at x 2 B on the interval x 20 the function f x has a global maximum at x 2 and a global minimum at x 0 C on the interval x 20 the function f x has no global minimum and no global maximum D on the interval z 20 the function f x has a global maximum at x 2 and no global minimum 5 5 points What is Ar for either a left or right hand sum with n 4 rectangles for 6 5 points The image below depicts which of the following approximations of 34 24 f x dx 12 1 f x da A A right hand sum with n 2 and Ax 6 B A left hand sum with n 6 and Ax 2 C A right hand sum with n 6 and Ax D A left hand sum with n 2 and Ax 6

8 5 points Use a calculator with definite integral capability to find decimal places A 2 4028 B 1040 0000 C 1037 7500 D 0 0039 E 0 7315 2 1 3 1 dr Give dr Give your answer t

PART 2 Show all your work for each opell answer without supporting work 11 15 points Consider the function f x 2x 3x 36x 4 a Find the critical point s of f x 2x 3x 36x14 6x 6x 136 Answer b Use either the first derivative test or the second derivative test the choice is yours to identify whether each critical point found in part a results in a local maximum or a local minimum for the original function f r Label each critical point as the location of a local maximum local minimum whichever is appropriate Answer c Use calculus to find the global maximum and global minimum of the function f x on the interval

4 5 points Which of the following is a true statement about the function f x graphed below 4 3 2 4 2 Y f b 3 X A on the interval 20 the function f x has a global minimum at z 3 and a global maximum at z 2 B on the interval z 20 the function f x has a global maximum at x 2 and a 0 global minimum at C on the interval z 20 the function f x has no global minimum and no global maximum D on the interval z 20 the function f x has a global maximum at x 2 and no global minimum

5 5 points What is Ar for either a left or right hand sum with n 4 rectangles for 34 2h A Az 8 5 B Az 4 C Az 8 D Az 34 E Ax 32 f x da

en y t 3t 6t 7 and y 0 12 find y t

13 5 points The rate r t upon the amount of time t in hours that has elapsed since they r t dt 0 035 JO What is the percent blood alcohol concentration of the adult 3 hours after finishing drinking if their blood alcohol concentration was 0 079 percent at the time they finished their last drink Include units in your answer Answer 14 5 points Consider the functions f x x x 7 and g x 8x 7 a Find the intersection points of these two functions by setting f x g x and solving for x Answer b Express the area trapped between these two functions as a definite integral and then use a calculator to evaluate this definite integral Give your answer to four decimal places

2 5 points A function f t has a local minimum at t 2 and a local maximum at t 4 Assuming t 2 and t 4 are critical points found by setting f t 0 and solving for t which of the following must be true Min t 2 210 max t 4 9 t 0 Af 2 is positive and f 4 is positive B f 2 is positive and f 4 is negative C f 2 is negative and f 4 is negative D f 2 is negative and f 4 is positive 4 420 242 0

1 5 points Which of the following is a critical point of the function f x ln x 8x A x 1 16 B x 16 fx ln x 8x C x 1 4 D z 4 E f x has no critical points

PART 1 For each multiple choice problem circle only one answer There is no partial credit for multiple choice problems 1 5 points Which of the following is a critical point of the function f x ln x 8x fx In x 8x A x 1 16 B 16 x 1 4 D x 4 E f x has no critical points 2 5 points A function f t has a local minimum at t 2 and a local maximum at t 4 Assuming t 2 and t 4 are critical points found by setting f t 0 and solving for t which of the following must be true Af 2 is positive and f 4 is positive B f 2 is positive and f 4 is negative C f 2 is negative and f 4 is negative D f 2 is negative and f 4 is positive t 2 210 max t 9 9 0 4 4 20 242 0 3 5 points Below is table of values for the second derivative of a smooth continuous function g x If x 3 and x 1 are the only candidates for where the original function g x may have an inflection point what can we conclude X 4 g x 2 3 0 1 2 0 4 0 20 A g x has an inflection point at both x 3 and x 1 B g x has an inflection point at x 3 but not at x 1 C g x has an inflection point at x 1 but not at x 3 D g x has no inflection points

Calculate the derivative of a function A r r when r 8

20 points We are given a vector field F x y z axy z Bx y where a are some nknown constants It is known that H identically and that 3 2k 2x x ind a and 3

3 Stem cells are a b C d e unspecialized specialized unable to divide always pluripotent only found in embryos

2 3 A No solution 7 6 5 4 3 2 10 B x 0 6 or x 2 1197 32 10 C All real numbers D x 46 4 or x 0 4 0 Find each quotient 2 VIAGIN A C 10 7 7 10 B D 34 35 10 7 A 15b 2b 20 C 15b 2b 8 5 789 3456 05 10 15 20 25 30 35 40 45 50 Simplify each difference 3 36 3b 1 86 7b 3 5b b 6b 3b 4 5b B 15b 2b 14 D 15b 2b 20 8h

2 Individuals with Pompe disease are unable to metabolize and are treated with a human enzyme produced by a glycogen transgenic hamster cells b lactose transgenic crops insulin cloned rabbits C d e cellulose transgenic hamster cells fats cloned hamster cells

Calculate the following expressions A Given h x x 2 x 4 h x

b mit Stolpirtia 1 Use the graph of g x shown to the right to answer the following questions A g 2 B lim g x C lim g x D g 3 y g x E List the values of x when g x is discontinuous 10 points

Find the volume V of the described solid S The base of S is the region enclosed by the parabola y 2 3x2 and the x axis Cross sections perpendicular to the y axis are squares V 8 X

Question 2 Solve this system of equations 4y x 4x No solution 4y 17 20

3 Solve this system of equations 4x 3y x y 29 2

2 f x x 6x 10 x Use the given zeros and the rational zeros t

function completely 3 f x x 2x 5x 6 Zero x 3 imions of which food danced in the children s heads as they slept in the poer Ju Refore Christmas

Twas The Night Bef 4 f x x 8x 11x 20 Zero x 4 Three of Santa s reindeer s names begin with the letter D What are those names SEAXE AD 10 MB 11 MA

1 f x 6x 2 x

lesha wants to prove that the measures of the 3 angles in a triangle always have a sum of 180 She decided to prove this conjecture using rigid motion She began with triangle ABC shown in Figure 1 B 16 Figure 1 C Figure 2 Rotate triangle ABC degrees on the midpoint of Rotate triangle ABC degrees on the midpoint of ince we used 2 rigid motions and all three angles form a straight line and a straight line equals angles will equal 180 degrees we can show that all

Explain what is meant when it is said that data vary How does the variability affect the results of statistical analysis Choose the correct answer below OA Saying data vary means that the values of the variable change every time they are observed Because data vary each statistical analysis of a given variable will lead to different results OB Saying data vary means that examining different types of subjects will yield different data such as examining a person and examining a car Because data vary it is impossible to compare different types of subjects such as comparing people and cars OC Saying data vary means that observations can be measured with fractions or decimals instead of only integers Because data vary it is important to make sure to accurately account for fractional observations OD Saying data vary means that the values of the variable change from individual to individual In addition certain variables can change over time for certain individuals Because data vary two different statistical analysis of the same variable can lead to different results

1 Use the definition of the derivative to show y f x x is f x 2x 2 Find the derivative f x if f x x sin x 3 Find the derivative f x if f x ln x x 1

4 Find the derivative f x if f x 11 m

9 points Find all local absloute maxima minima of the function f x 2x 1 1 x 5 on the real line o

3 Given the graph of the function f a below answer the following questions 50 40 30 20 10 2 3 Hin y f x 5 a Is f x one to one Explain b Use the graph to approximate f 20 c Use the graph to approximate f 20 x

6 Find the anti derivative sin 2 cos x dx