Indefinite Integration Questions and Answers

1 point The graph of f is shown below Evaluate each integral by interpreting it in terms of areas 1 1 2 dz 0 f x 2 f x dx 0 3 ff x dz 0

Solve the following equation 11 30 1 35 424 Write your answers exactly i e no decimals and simplified if possible separated by a comma Hint X Write your answers as decimals rounded to four decimal places

Use the like bases property to solve the equation n 1 36n 10 If needed enter a fraction value not a decimal value n

ans 1 A vehicle purchased for 22400 depreciates at a constant rate of 4 Determine the approximate value of the vehicle 10 years after purchase Round to the nearest whole number

Evaluate the integral using any appropriate algebraic method or trigonometric identity dy 56 y 1 y2 7 S dy 56 y 1 y2 7

Homework C156 Question 8 of 9 1 point Question Attemp Quadratic function f x x graph is a parabola X 2 1 0 1 2 9 8 f x 4 1 0 1 4 Bar es G 3 V

S lve for the unknown quantities in triangle ABC Round values to the nearest tenth A 11 2 131 6 C B 97 7 ft OA B 37 2 b 376 1 ft c 304 1 ft OB B 37 2 b 31 4 ft c 25 5 ft OC B 36 8 b 301 3 ft c 376 1 ft OD B 37 2 b 304 1 ft c 376 1 ft

Evaluate the integral 3 L x 1 dx x

Solve the triangle ABC Find the unknown side b the side opposite angle B b Round to the nearest tenth as needed 10 5 B 106 14 5

Evaluate the indefinite integral x 5 S x 10x 2 dx

Evaluate the indefinite integral Sz COS X 4 sin x 16 dx

Evaluate the indefinite integral ln x 6 X js dx

Evaluate the definite integral 11 4 3 x 4 dx

For the given function f x find the derivative f 1 a at the given point a first finding b f a f x 7x59x 24x a 22 b a

x 4x 16x 11 dx 0 X X 123 8x 64x dx

6 7 I c cos lnx dx ev dx

t cosh t dt 5 In an den

1 3 arctan 1 r dr It cosht dt

tegration Workshe 1 xnx x ln x dx

al the Taylor rals xercises revolution and evaluate the integral using appropriate technology If not possible explain why 8 Consider the curves given by y sin x and y cos x For each of the following problems you should include a sketch of the region solid being considered as well as a labeled representative slice a Sketch the region R bounded by the y axis and the curves y sin x and y cos x up to the first positive value of x at which they intersect What is the exact intersection point of the curves b Set up a definite integral whose value is the exact area of R c Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving R about the z axis d Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving R about the y axis e Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving R about the line y 2 f Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving R about the line x 1

Consider the two integrals below a 00 14 28 dr 1 x8 b 11 1 112 3 da 2 Classify the integrals above as an improper integral or a regular definite integral State WHY the integra improper if it is Justify your classification

A population of protozoa develops with a constant relative growth rate of 0 8877 per member per day On day zero the population consists of 4 members Find the population size after 4 days Round your answer to the nearest whole number P 4

Let f x x a Find the tangent line to the graph of the function at x 11 help formulas y b Find the normal line to the graph of the function at x 11 help formulas y

Subtract State the difference in simplest form 24x 0 4x x 3 8x 24 0 4x43 3 0 x 3 fr dr vt

10 a Define the inverse sine function the inverse cosine func tion and the inverse tangent function cos 1 V2 and tan 1 b Find sin 1 c For what values of x is the equation sin sin x x true For what values of x is the equation sin sin x x true

Find the following integral ex e x Sex dx ex e x ex e x dx

2 pts Let fR R be differentiable We say that f is convex if for all x y R and all t 0 1 f tx 1 t y tf x 1 t f y i e that the graph of f is below its chords a 1 pt Prove that f is convex if and only if f y f x Vf x y x for all r y R i e the graph of f is above its tangent lines b 0 5 pt Assume that f is C Prove that if f is convex then n fax x h h 20 i j 1 for all x h ER We recall that for a function g R R twice differentiable we have the following Taylor formula for all a b E R there exists c a b such that g c 2 This is no longer true for functions of several variables c 0 5 pt Applying the previous formula to the function g t f x th between a 0 and b 1 prove the reverse implication in the g a g b g b a b

1 1 pt Sketch the spiral r 0 sin 0 0 0 you may use a screenshot of Desmos if you like a 4 pts Determine the exact area that would be enclosed in the portion of the curve in quadrant c

Area of a triangle Given a triangle that has vertices x y x y2 and x3 y3 the area enclosed by the 1 1 1 triangle is given by A IDI where D Determine the area of the triangle X1 X2 X3 V V2 V3 with vertices given by 1 6 6 2 and 6 8 where units of length are measured in inches 8 pts

Propion K When Greg s father was in high s for the 100 m race Which distan

Find the domain and range of the function Enter your answers using interval notation f x 4x 2 domain range

Evaluate Stan xdx a c tan x 3 tan x x C tan x 3 tan x x C b d tan x tan x x tan 3 X tan x x

2 a Evaluate In x dx x lnx 2x lnx 2x C x ln x 2x ln x 2x C b d x lnx 2x ln x 2x C 2x ln x 2x C

Evaluate a c 3 2 3 2 X cos sin 2 Cos 2 sin C 3 X s sin sin C 2 2 3 2 b d 2 3 cos x 4 sin sin 2 3 2 x 2 cos sin sin 3 2 23

Express the function or rule in words g x x 6 5 O Add 6 then divide by 5 O Square then add 5 Divide by 5 then add 6 O Multiply by 5 then subtract 6 O Add 6 then multiply by 5

5 a c Evaluate fx sin xdx x cos x 2xsin x 2 cos x C x cos x 2x sin x 2 cos x C b d x cos x 2x sin x 2 cos x C x cos x 2x sin x 2 cos x

STATINE TONNA NAS SANA The region bounded by the given curves is rotated about the specified axis Find the volume of the resulting solid by any method y x 23x 132 y 0 about the y axis BINNENNA HANTARA CONTA NNNN UNNURE SNINGAE CHE O 2 SAD WWWANAH Sek FRENAN S G A A INGRIMEY

Evaluate fx e dx a c e x 3x 6x 6 C e 3x x x 6 C d b e x 3x 6x 6 e x 3x x 6 C

D 3 Aita 2 Show Transcribed Text D D 3 2 1 3 2 3 4 5 C 1 2 3 4 5 f local max and min fu Even and add I a Find the domain of F 010 b Find the range C Determine F 0 D X intercept smallest to largest e Decreasing

a c tan x 3 tan x 3 tan x x C tan x x C b d tan x tan x x C tan x 3 tan x x C

5 Evaluate cos s dx a 2 cos sin 4 3 E C b C d 2 3 1 3 cos sin cos sin sin

3 cos 2x cos 3x dx

Evaluate x e dx a c e x 3x 6x 6 C e 3x x x 6 C b d e x 3x 6x 6 C e x 3x x 6 C

Evaluate sinh 2ln3 a 40 9 c 9 b d 41 9 0

4 tegrals f sin x cos x dx

rove the identity cosh x sinh x ex cosh x sinh x 2 1 ex e x 1 2 1

n investment worth 1 million in 2005 has been growing at a rate of f t 0 141 1 12 million dollars per year here t is the number of years since 2005 a Calculate how much the investment will have grown between 2005 and 2012 Round your answer to three decimal places million How much is it projected to grow between 2012 and 2017 Round your answer to three decimal places million b Recover the function for the model that gives future value of an investment in million dollars t years since 2005 The coefficient of integration should be roun three decimal places F t million dollars

1 For what values of a b and c is the integral a rational function 2 Evaluate the indefinite integral 3 Evaluate the definite integral L bx cdx 2 x 1 2 2 r x 1 dr dr 2 2x 2

Problem 5 Evaluate the following integrals using trigononometric substitutions ez a 2 S dx b 1 1 da Hint you can use f sec x Intan x sec x C and f sec x dx sec z tan x In sec x tan x C 3 dx c d fx x 9dx

tition is allowed b if no repeats are allowed c Find the number of combinations of 2 elements taken 2 at a time Does this answer differ from that in par b a If repetition is allowed how many ways can 2 letters be chosen from the set P Q if order is important Ch the correct formula below and if necessary fill in the answer boxes to complete your choice Type the terms of your expression in the same order as they appear in the original expression A 2P BM 2 How many ways can 2 letters be chosen from the set P Q if order is important 4 b If no repeats are allowed how many ways can 2 letters be chosen from the set P Q if order is important the correct formula below and if necessary fill in the answer boxes to complete your choice Type the terms of your expression in the same order as they appear in the original expression OA M 21 2 2 How many ways can 2 letters be chosen from the set P Q if order is important c Find the number of combinations of 2 elements taken 2 at a time Choose the correct formula below and necessary fill in the answer boxes to complete your choice Type the terms of your expression in the same order as they appear in the original expression OA M B 21 2 2 2 Find the number of combinations of 2 elements taken 2 at a time