Calculus Questions

Evaluate the definite integral 5 6 v1 3x dx

2 1 1 M W Sketch the region enclosed by the given curves Decide whether to integrate with respect to x or y Draw a typical approximating rectangle 2x y 8 x y 4 4 al JS Secti 2 1 al WebAssign Plot ah 1 2

Evaluate the definite integral 8 X

O The matrix is in RRE form O There is a leading entry of a row that is NOT one 1 0 3 0 0 0 0 1 4 O Leading ones DO NOT appear further to the right from row to row O O There is a row of zeros that is NOT at the bottom of the matrix Matrix 4 4 0 1 NOT all other entries in a column with a leading one are zero The matrix is in RRE form 1 3 0 4 004 4 000 0 There is a leading entry of a row that is NOT one

Solve the system below by using Gauss Jordan Elimination This means convert the system to an augmented matrix put that matrix in reduced row echelon RRE form using row operations and determine the solution x y 5z 2 y 4z 1 2x7y 22z 11 a Enter the RRE form of the augmented matrix RRE form of augmented matrix Give the solution to the system If there is a unique solution enter it as an ordered triplet in the form x y z If there is no solution enter Inconsistent If there are free variables enter your answer as an ordered triplet in terms of the free variables

2 Get a formula for the derivative of the following functions f x 2x 5x 3x 1 k x x 3x 2 x 1 g x m x x 1 tan x x 1 h x n x 1 x 5 3x cos x

s x sin x cos x v x sin x X t x w x X cos x sin x u x x 4x 5x x 3x 2 y x 2x 5x 1

p x x 4 sin x s x sin x cos x sin x v x X x 2 tan 3x 1 q x t x w x X cos x sin x r x 2 2 x 2 x 4x 5x x 3x 2 u x y x 2x 5x 1

Are the following functions derivable at the given point If so calculate the derivative f x g x x if x 0 200 if x 0 sin x x in x if x 0 if x 0 0 in x 0

nd the domain and range of the function 6 1 g z 2 1 A D 1 00 R 0 00 C D 1 R 0 00 B D 1 0 R 0 D D 0 R 0 00

Calculate the second and the third derivatives of the following functions f x 4x 6x 2x 1 k x sin 2x x 1 g x x 1 h x cos 2x x 2x 1 m x 2 n x x 4x x 1

A chemist has a 100 gram sample of a radioactive material He records the amount of radioactive material every week for 6 weeks and obtains the following data Complete parts a f Week x 0 1 2 3 4 5 6 Weight y in grams 100 0 82 7 66 5 56 9 45 3 37 2 30 5 a Treating week x as the independent variable use a graphing utility to fit an expone function to the data y ab Round to four decimal places as needed

Calculate the following derivatives using the definition f x 3x on x 3 g x 4x on x 1 h x 2x 5x on x 1 k x 4x on x 2

The function f x x5 52x 2x 147x 98 has the graph given to the right Use the graph to factor the polynomial What is the factored form of the polynomial f x Simplify your answer Type your answer in factored form Factor completely 10 6000 12000 10 O G

A person wants to purchase both parcels of land shown in the figure to the right and make a check on their combined area There is no equation for the river frontage so the person uses the average of the left sum of rectangles covering the area The 1 000 foot baseline is divided into 10 equal parts At the end of each subinterval a measurement is made from the baseline to the river and the results are tabulated Let x be the distance from the left end of the baseline and let h x be the distance from the baseline to the river at x Use L 0 to estimate the combined area of both parcels and calculate an error bound for this estimate How many subdivisions of the baseline would be required so that the error incurred in using L would not exceed 2 500 square feet X 0 h x 0 700 200 100 300 400 500 600 183 235 240 324 390 255 284 The combined area of both parcels is 1 ft 800 445 900 480 River Parcel 1 1 000 ft 1 000 500 Parcel 2 h x 500

3 2 6 5 32 2 2 3 5 6 At the point shown on the function above which of the following is true Of 0 f 0 of 0 f 0 Of 0 f 0 f 0 f 0

Suppose parametric equations for the line segment between 7 6 and 4 3 have the form a b If the parametric curve starts at 7 6 when t 0 and ends at 4 3 at t 1 then find a b c and d d X Y a bt c dt

Find the standard form of the equation of the ellipse with the given characteristics Center 2 9 a 3c foci 5 9 9 9

42Y BIOMEDICAL CHOLESTEROL An experimental drug lowers a patient s blood serum cholesterol at the rate of t 25 t2 units per day where t is the number of days since the drug was administered 0 t 5 Find the total change during the first 3 days

The ellipse x 4 6 1 can be drawn with parametric equations Assume the curve is traced clockwise as the parameter increases If x 4 cos t

Find the center vertices and foci of the hyperbola y 5 x 3 1 25 center x y vertices x y foci x y x y x y 1 8 smaller y value larger y value 2 smaller y value Find the asymptotes of the hyperbola Enter your answers as a comma separated list of equations larger y value Sketch its graph using the asymptotes as an aid Use a graphing utility to verify your graph y y 10

Express x eliminating ln 3t y 2 t in the form y f x by the parameter y

The filament of an automobile headlight is at the focus of a parabolic reflector which sends light out in a straight beam see figure 8 in 1 4 in a The filament of the headlight is 1 4 inches from the vertex Find an equation for the cross section of the reflector Assume that the vertex i inches b The reflector is 8 inches wide Find the depth of the reflector Round your answer to two decimal places in

veyor measurements found below Find the distance across the lake Round your answer to the nearest tenth Draw a snowflake next to your name for 1 point 85 ft 62 120 ft

Calculate the indicated Riemann sum S5 for the function f x 26 2x Partition 2 8 into five subintervals of equal length and for each subinterval XK 1 xk let CK S5 let Ck xk 1 Xk 2

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin 10 4 0 5 y 10 5 5 0 3 2 4 0 0 3 2 5 10 X

Write the first five terms of the sequence defined recursively a 27 ak 1 ak 4 a1 a a3 34 a5 Use the pattern to write the nth term of the sequence as a function of n Assume that n begins with 1 an

a Given that u 2 5 and 1 3 find the new vector w 2 v 5 9b What is the component form of

8 Consider the vector whose initial point is P 3 2 and terminal point is Q 5 1 Find the position vector and graph both vectors in the space provided

Solve the triangle rounding each nswer to the nearest tenth as needed B a 5 7 b 4 3 59 Y

In this question we will use Newton s Method to approximate the x value of the intersection between f x and g x closest to 2 Start your approximation with co 2 and apply two iterations of Newton s Method to the function F x f x g x Give your answers correct to at least 4 decimal places Remember that WeBWork can understand calculator ready numbers and make liberal use of copy paste 21

In this question we will use a spreadsheet to approximate the root of f x x5 3x 9 using Newton s method with 1 The picture below shows a spreadsheet being used to perform the calculation A downwards arrow indicates the contents from the cell above will be copied down a Which entry should go in cell A1 O 5 B1 4 9 B1 2 A1 5 3 A1 3 9 00 5 A1 4 9 A1 2 O B1 5 3 B1 3 9 01 b Which entry should go in cell B1 00 O A1 5 3 A1 3 9 01 O 5 B1 4 9 B1 2 O B1 5 3 B1 3 9 O 5 A1 4 9 A1 2 c Which entry should go in cell C1 O A1 5 3 A1 3 9 O B1 5 3 B1 3 9 O 5 B1 4 9 B1 2 A 2 A1 B1 C1 1234 B C

Write the first five terms of the sequence Assume n begins with 1 an 110 9n a 101 32 92 a3 83 34 74 35 65 Determine whether or not the sequence is arithmetic If it is find the common difference d If the sequence is not arithmetic enter NOR Yes the sequence is arithmetic O No the sequence is not arithmetic X

Find the standard form of the equation of the parabola with the given characteristic s and vertex at the origin Focus 2 0

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin Foci 7 0 major axis of length 18

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin 10 4 0 5 y 10 0 7 2 4 0 5 0 7 2 10 X

Use the Binomial Theorem to expand and simplify the expression x 1 5

Calculate the binomial coefficient 100 2

Given u x t t e compute

Find all the first and second order partial derivatives of f x y 8 sin 2x y 10 cos x y A f fx B af fy 8cos 2x y 10sin x y c f fxx E D fay dady F r 1 32sin 2x y 10cos x y 8sin 2x y 10cos x y 16sin 2x y 10cos x y 16sin 2x y 10cos x y B

Consider the concentration C in mg litre of a drug in the blood as a function of the amount of drug injected x in mg and the time since injection t in hours For 0 x 4 mg and t 0 hours we have C f x t 26xte 4 Calculate the following f 3 4 0 00003511 Give a practical interpretation of your answer Choose the best option f 3 4 is A the amount of a 4 mg dose in the blood 3 hours before injection B the change in concentration of a 4 mg dose in the blood 3 hours after injection C the concentration in the blood 4 hours after injection of a 3 mg dose D the half life of a 3 mg dose in the blood 4 hours after injection E the concentration in the blood 3 hours after injection of a 4 mg dose O F the change in concentration of a 3 mg dose in the blood 4 hours after injection

Suppose f x y is a differentiable function Match the names of its derivatives in Newtonian and Leibnitz notation 1 2 A 3 81 5 82 Dy Of 4 8x dydn A f B fam c fry D fy E fym F fyy

Consider the following function f x 1622 3 Find f 64 and f 64 f 64 64 Find all values c in 64 64 such that f c 0 Enter your answers as a comma separated list If a answer does not exist enter DNE C Based off of this information what conclusions can be made about Rolle s Theorem O This contradicts Rolle s Theorem since f 64 f 64 there should exist a number c in 64 64 such that f c 0 This does not contradict Rolle s Theorem since f 0 does not exist and so f is not differential on 64 64 O This does not contradict Rolle s Theorem since f 0 0 and 0 is in the interval 64 64 O Nothing can be concluded This contradicts Rolle s Theorem since f is differentiable f 64 f 64 and f c 0 e but c is not in 64 64

Score 3 9 3 9 answered Question 3 C 3 Save progress Done 0 1 pt 6 Deta Verify that the function satisfies the three hypotheses of Rolle s Theorem on the given interval Then find all numbers c that satisfy the conclusion of Rolle s Theorem Enter your answers as a comma separated list f x cos 52 20 7 20

Score 2 9 279 answered Question 8 f 5 Suppose that 4 f x 5 for all values of x What are the minimum and maximum possible values of 5 f 3 0 1 pt 6 f 3

Question 7 If 1 5 and f x 1 for 1 x 4 how small can f 4 possibly be

Calculate the indicated Riemann sum S3 for the function f x x 4x 5 Partition 0 3 into three subintervals of equal length and let c 0 7 c 1 3 and c3 2 8 S3 Simplify your answer

Compute L3 and R3 for Graphs A and B The value of L3 for Graph A is Simplify your answer AY 16 14 12 10 86 4 2 04 0 Ay 16 14 12 10 8 6 4 2 y g 4 6 Graph A 8 x 10 Q E Q

Find the partial sum without using a graphing utility 100 50 En En n 51 n 1

Use the Binomial Theorem to expand and simplify the expression x 1 5