Calculus Questions

T 2T Find the volume of the portion of the solid sphere p 2 that lies between the cones 3 and 3 The volume is Type an exact answer using and radicals as needed

1 How many significant figures are there in following number 3 006 Answer 2 How many significant figures are there in following number 0 009 Answer 3 Round the following number to three significant figures 0 08707 Answer 4 Round the following number to three significant figures 12 18 Answer 5 Round the following number to three significant figures 252 874

Consider the equation y 1 2 cos 4x Give all answer in exact form NO DECIMALS Represent phase shift and vertical shift with a signed number to show direction and distance n teft n right 0 What is the amplitude 1 What is the period length 2 How far apart are the critical points 3 What is the phase shift 4 What is the vertical shift 5 What is the equation for for midline

a x y a u v a Solve the system u 2x 3y v x 5y for x and y in terms of u and v Then find the value of the Jacobian b Find the image under the transformation u 2x 3y v x 5y of the triangular region in the xy plane bounded by the x axis the y axis and the line x y 2 Sketch the transformed image in the uv pla a x y

0 2 4 3 0 2 SS S p sin 24 do d0 dp 2 0 2x 4

Ketch the graph described by the following cylindrical coordinates in three dimensional space 1 7 Choose the correct answer below A OE OB COLLE O C OG 246 OD 14 OH

A 23 ft ladder leans against a building so that the angle between the ground and the ladder is 75 How high does the ladder reach on the building ft Report the exact answer or use the col

Solve the following trigonometric equations without using a calculator Find all solutions where 0 x 2 If no solution exists type DNE 2 cos x 1 0 Solution Select an answer 18 Answer in RADIANS

A 9 foot ladder is placed against a building If the base of the ladder is 4 feet away from the building how far up the building will the ladder reach Round the answer to the nearest tenth X

Find the length of an arc if the radius of the arc is 1 7 cm and the measure of the arc is 1 6 radians cm round answer to three decimal places

lim f x 0 x a lim g x 0 x a lim h x 1 x a lim p x x a lim q x x a evaluate if the following limits not are indeterminate forms If a limit is indeterminate enter INDETERMINATE a lim f x 9 x x a f b lim f x P x x a c lim h x P x x a d lim p x f x x a e lim p x 9 x x a lim x p x x a

Find the derivative dx 4 X d 4 5 5 X

Differentiate P 0 y 10 X X 10 dx 10 10 X X 11

Compute the mean and standard deviation of the provided sample data on cost of living adjustments Then compute the z scores for the smallest and largest comparative salaries and housing adjustments and interpret your results Click the icon to view the data UCTIQUIVII Find the largest and smallest comparative salary amounts and calculate the z scores for these amounts Round to two decimal places as needed City Comparative Salary 54 578 136 024 Columbus New York Housing 9 479 Z scores Calculate the z scores for the largest and smallest housing percentages Round to two decimal places as needed City St Louis New York 0 80 2 77 is the farthest away from the Z scores 0 75 2 75 Interpret your results The z scores show that the comparative salary amount for is the farthest away from the comparative salary amount The z scores also show that the housing amount for housing amount 3 San Francisco 4 Columbus 5 Seattle 6 Louisville 7 Austin 8 Philadelphia 9 St Louis 10 Pittsburgh 11 Charleston 12 Dallas 13 Charlotte 14 DC 15 New York 16 Portland 17 18 19 20 21 22 23 24 25 26 27 28 29 105241 54578 83253 55602 57530 72048 56084 59578 60904 58072 58012 87892 136024 77349 304 1 133 2 13 72 21 22 4 5 214 479 108

Find the linearization L x y of the function f x y 2x 2y 4 at each point a 0 0 b 1 2 a L x y Type an exact answer using as needed

By about how much will g x y z x x cos z y sinz y change if the point P x y z moves from Po 1 1 0 a distance of ds 0 1 unit toward the point P 1 1 2 dg Type an integer or decimal rounded to four decimal places as needed

Find the linearization L x y of the function f x y e cos y at the points 0 0 and 0 0 7 The linearization at 0 0 is L x y Type an exact answer using as needed COO

5 Another set of conditions that will prove that a quadrilateral is a rectangle is the following Any parallelogram with at least one right angle is a rectangle Given EFGH is a parallelogram with ZH being a right angle Prove EFGH is a rectangle use of a paragraph proof here is nice E

Find the directions in which the function increases and decreases most rapidly at Po Then find the derivatives of the function in these directions f x y z x y yz Po 1 1 4 5 27 j The direction in which the given function f x y z x y yz increases most rapidly at Po 1 1 4 is u 27 Type exact answers using radicals as needed The direction in which the given function f x y z x y yz decreases most rapidly at Po 1 1 4 is v k Type exact answers using radicals as needed 27

28 Macmillan Learning a Suppose a researcher runs a t test of the difference of two independent means The samples are taken from populations of unknown variances that are not assumed to be equal The sample sizes and standard deviations are summarized in the table Sample Group Sample size m 22 m 47 df standard deviation 83 8742 32 5 7356 Use software to compute the Welch Satterthwaite approximation of the degrees of freedom df You might find some software manuals useful Give your answer precise to at least three decimal places Attemp

Using the accompanying Weddings data find the mean median and standard deviation of the wedding costs What would you tell a newly engaged couple about what cost to expect Consider the effect of possible outliers in the data Click the icon to view the Weddings data For the given data set the mean of the wedding costs is 25772 The median of the wedding costs is 24400 The standard deviation of the wedding costs is 13205 Type integers or decimals rounded to two decimal places as needed mild outlier s for the costs Consider the effect of possible outliers in the data There is are Remove these outliers and recalculate the mean median and standard deviation Recalculating these statistics the mean of the wedding costs is The median the wedding costs is The standard deviation of the wedding costs is Type integers or decimals rounded to two decimal places a needed 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 53000 46400 41800 33600 30100 29800 29300 27200 27000 24800 24600 24400 21400 20300 20300 20300 18600 18400 18100 15400 14400 13000 6000

hat contains 4 marbles one inge one a Determine the number of sample points in the sample space There are sample points in the sample space b Construct a tree diagram and list the sample space Construct a tree diagram Choose the correct answer below O O een one red and Click here to view diagram d Click here to view diagram b Click here to view diagram c Click here to view diagram a List the sample space Choose the correct answer below OA OG OR OY GO GR GY RO RG RY YO YG YR B O G R Y C 00 OG OR OY GO GG GR GY RO RG RR RY YO YG YR YY D OO GG RR YY c Determine the probability of selecting exactly 1 green marble The probability is Simplify your answer d Determine the probability of selecting at least 1 marble that is not green

Two coins are tossed Assume that each event is equally likely to occur a Use the counting principle to determine the number of sample points in the sample space b Construct a tree diagram and list the sample space c Determine the probability that no tails are tossed d Determine the probability that exactly one tail is tossed e Determine the probability that two tails are tossed 1 Determine the probability that at least one tail is tossed a There is are sample point s in the sample space b Construct a tree diagram Choose the correct answer below OA OB What is the sample space Choose the correct answer below OA H T HH HT TH TT OB H T H T OC H T D HH HT TH TT c The probability that no tails are tossed is Simplify your answer d The probability that exactly one tail is tossed is Simplify your answer Simplify your answer e The probability that two tails are tossed is 1 The probability that at least one tail is tossed is Simplify your answer GECER OC OD IK

For the data provided complete the following tasks a Apply both column and line sparklines to visualize the trends in the prices for each of the four stocks b Compute the daily change in the Dow Jones index and apply a win loss sparkline to visualize the daily up or down movement in the index Click the icon to view the closing stock prices Choose the correct column and line sparklines for GE O A Q Q OB Q O C Q Q 5 O D o 3 9 8 2010 4 9 9 2010 5 9 10 2010 6 9 13 2010 7 9 14 2010 8 9 15 2010 9 9 16 2010 10 9 17 2010 11 9 20 2010 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 126 08 126 36 127 99 129 61 128 85 129 43 129 67 130 19 131 79 17 9 18 17 97 18 56 18 74 18 72 18 97 18 81 18 93 20 64 20 61 20 62 21 26 21 45 21 59 21 93 21 86 21 75 15 7 15 91 15 98 16 25 16 16 16 34 16 23 16 29 16 55

Determine the number of terms required to approximate the sum of the convergent series with an error less than 0 001 1 1 6n n 1

Use Lagrange Multipliers to find the maximum and minimum values of f x y z x 3y ubject to the constraint x 4y 2 36

Write an expression for the nth partial sum of the series Using the nth partial sum and the Definition of Convergence Divergence of a Series determine whether the series converges or diverges If convergent what is the sum of the series n 2 n Converges n

Find the nth term that generates the sequence then determine whether the sequence converges or diverges 1 8 27 64 25 50 75 100

Show your work as you determine whether the sequence or series converges or diverges 8 1 n 1 O Converges O Diverges 2n 3 3n 4

Determine if convergent or divergent If convergent state whether conditionally convergent or absolutely convergent 1 n 1 O Divergent 1 n Conditionally Convergent Absolutely Convergent

Sketch several solution curves in the phase plane of the system of differential equations dx dt Ax using the given eigenvalues and eigenvectors of A A 2 A 4 V H V2 A

Select all expressions that are equivalent to 2 10 A 2 9 2 B 2 11 2 C 10 2 D 10 2 E 2 5 2

ts 3 Consider the curve C defined by the polar equation r 2 sin 30 1 Set up the area A of the region bounded by the curve C depicted below DO NOT EVALUATE THE DEFINITE INTEGRAL

Find the limit Use l Hospital s Rule where appropriate If there is a more elementary method consider using it 2x 8 X 4 lim X 4

5 4 3 2 1 1 2 3 4 5 6 a Find the interval s on which f is increasing Enter your answer using interval notation 1 3 U 4 6 b Find the interval s on which f is decreasing Enter your answer using interval notation 0 1 U 3 4 c Find the open interval s on which f is concave upward Enter your answer using interval notation 0 2 d Find the interval s on which fis concave downward Enter your answer using interval notation 2 4 U 4 6 e Find the coordinates of the point s of inflection

Since the derivative of sin x is cos x and the derivative of 96x is 96 applying L Hospital s Rule a fourth time gives us the following sin x lim x 0 96x lim cos x X 0 96 Therefore we have the following 1 2x lim X 0 cos x 1 4x4 lim x 0 result sin x 16x X lim X 0 1 cos x 48x sin x lim x 0 96x lim X 0 cos x 96

After applying L Hospital s Rule three times we have lim sin x So we still have an indeterminate limit of type x 0 96x find additional derivatives The derivative of sin x with respect to x is The derivative of 96x with respect to x is We will apply L Hospital s Rule for a fourth time To do so we need to

After applying L Hospital s Rule once we have lim 8 sec x 1 x0 x sec x tan x needed and we will do so here applying the rule for a second time To do so we need to find additional derivatives The derivative of 8 sec x 1 with respect to x is The derivative of x sec x tan x with respect to x is So we still have an indeterminate limit of type o lo We recall that L Hospital s Rule can be applied repeatedly as

Since the derivative of x sin x is 1 cos x and the derivative of 16x3 is 48x2 applying L Hospital s Rule a second time gives us the followin 1 cos x 48x lim x 0 x sin x 16x lim x 0 Analyzing this we see that as x 0 1 cos x and 48x

x sin x 16x3 we will do so here applying the rule for a second time To do so we need to find additional derivatives The derivative of x sin x with respect to x is After applying L Hospital s Rule once we have lim x 0 The derivative of 16x3 with respect to x is So we still have an indeterminate limit of type olo 0 We recall that L Hospital s Rule can be applied repeatedly as needed and

Since f x 8 sec x 1 and g x x sec x tan x applying L Hospital s Rule gives us the following f x lim x a g x lim x 0 lim x a g x 8 tan x x x tan x 8 sec x 1 lim x ot x sec x tan x Analyzing this we see that as x 0 8 sec x 1 and x sec x tan x

Since f x x sin x and g x 16x3 applying L Hospital s Rule gives us the following f x lim x a g x lim x 0 f x lim x a g x x 2 cos x 1 4x4 lim x sin x x 0 16x Analyzing this we see that as x 0 x sin x and 16x

Consider the following graph 5 4 3 2 y 1 1 2 3 4 5 6 X a Find the interval s on which f is increasing Enter your answer using interval notation 0 1 U 3 7 b Find the interval s on which f is decreasing Enter your answer using interval notation 1 3 X c Find the interval s on which f is concave upward Enter your answer using interval notation 2 4 U 5 7

K Find the area under the standard normal curve to the right of z 1 25 OA 0 5843 OB 0 7193 OC 0 6978 OD 0 8944

Recall that in general if we have a limit of the following form where both f x 0 and g x 0 then the limit may or may not exist and is called an indeterminate form of type 0 0 We note that this type of limit can be evaluated for certain functions by dividing numerator and denominator by the highest power of x that occurs in the denominator However we are give the following limit where this method does not work f x lim x a g x f x lim x10 For the evaluation of indeterminate forms of type ew where other methods do not work we use L Hospital s Rule This rule states that if f and g are differentiable and g x 0 on an open interval I that contains a except possibly at a where lim f x 0 and lim g x 0 then the following holds as long as the limit on the right side exists or is or x a x a g x cos x 1 x 4x4 We can apply L Hospital s Rule for the given function using a 0 f x cos x 1 1x f x lim lim x a g x x a g x 16x x and g x 4x4 We need to first find both f x and g x

As x 0 8 tan x x f and x tan x

We first rewrite the given limit such that we have a single rational expression as follows 8 lim x 0 X 8 tan x lim x 0 x tan x 022

The graph of the derivative f of a continuous function f is shown Assume f continues to 4 2 2 X y y f x 4 X 6 8 a On what interval s is f increasing Enter your answer in interval notation X X On what interval s is f decreasing Enter your answer in interval notation Your answer cannot be understood or graded More Information b At what value s of x does f have a local maximum Enter your answers as a comma separated list X At what value s of x does f have a local minimum Enter your answers as a comma separated list

Suppose that 11 000 is invested at continuous rater an grows to 81 108 in 22 years Find the interest rate r The interest rate is Round to the nearest percent as needed

What is the algebraic expre x x 5 Do not simplify