Mth 231 written assignment 5. MODULE 10 WRITTEN ASSGINEMNT 5 ONLY QUESTIONS #1-20PLEASE TYPE ANSWER ON WORD DOCUMENT PROVIDED IN MATH LAB OR MATH TYPE. LEAVE SPACE AFTER ANSWER FOR COMMENTS.PLEASE SHOW ALL WORK FOR ANSWER.I HAVE ATTACHED TWO COPIES.WORD DOCUMENT IS TO BE USED TO TYPE ANSWERS UNDER EACH QUESTION AND PDF IS A REFERENCE WITH ANSWERS ATTACHED.
MAN-231: CALCULUS I WRITTEN ASSIGNMENTS NAVIGATION Written Assignments Module 2 Written Assignment 1 Module 4 Written Assignment 2 Module 6 Written Assignment 3 Module 8 Written Assignment 4 Module 10 Written Assignment 5 Module 12 Written Assignment 6 Module 2—Written Assignment 1 For #1 and #2 - Find the domain, range, and all zeros/intercepts, if any, of the function. 1. 2. 3. For the functions Find: a. f + g - Determine the domain of the new function b. f − g - Determine the domain of the new function c. f · g - Determine the domain of the new function d. f /g - Determine the domain of the new function 4. A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by where t is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture. a. Express the area of the bacteria as a function of time. b. Find the exact and approximate area of the bacterial culture in 3 hours. c. Express the circumference of the bacteria as a function of time. d. Find the exact and approximate circumference of the bacteria in 3 hours. 5. Write the equation of the line satisfying the given conditions in slope-intercept form. 6. Write the equation of the line satisfying the given conditions in slope-intercept form. 7. A house purchased for $250,000 is expected to be worth twice its purchase price in 18 years. a. Find a linear function that models the price P of the house versus the number of years t since the original purchase. b. Interpret the slope of the graph of P. c. Find the price of the house 15 years from when it was originally purchased. 8. Consider triangle ABC, a right triangle with a right angle at C. a. Find the missing side of the triangle. b. Find the six trigonometric function values for the angle at A. Where necessary, round to one decimal place. 9. P is a point on the unit circle. a. Find the (exact) missing coordinate value of each point b. Find the values of the six trigonometric functions for the angle θ with a terminal side that passes through point P. Rationalize denominators. 10. Solve the trigonometric equations on the interval 0 ≤ θ < 2π. 11. find a. the amplitude, b. the period, and c. the phase shift with direction for each function. 12. a. find the inverse function, and b. find the domain and range of the inverse function. 13. evaluate the function. give the exact value. 14. the depth (in feet) of water at a dock changes with the rise and fall of tides. it is modeled by the function where t is the number of hours after midnight. determine the first time after midnight when the depth is 11.75 ft. 15. write the equation in equivalent exponential form. 16. write the equation in equivalent logarithmic form. for #17 and #18 - use properties of logarithms to write the expressions as a sum, difference, and/or product of logarithms. 17. 18. for # 19 and # 20 - use the change-of-base formula and either base 10 or base e to evaluate the given expressions. answer in exact form and in approximate form, rounding to four decimal places. 19. 20. module 4—written assignment 2 1. points p(4, 2) and q(x, y) are on the graph of the function f (x) = . complete the following table with the appropriate values: y-coordinate of q, the point q(x, y), and the slope of the secant line passing through points p and q. round your answer to eight significant digits. 2. use the value in the preceding exercise to find the equation of the tangent line at point p. 3. consider the function f (x) = −x2 + 1 sketch the graph of f over the interval [−1, 1]. 4. approximate the area of the region between the x-axis and the graph of f over the interval [−1, 1] for problem 3. 5. use the graph of the function y = f (x) shown here to find the values of the four limits, if possible. estimate when necessary. 6. use direct substitution to show that each limit leads to the indeterminate form 0/0. then, evaluate the limit. 7. use direct substitution to show that each limit leads to the indeterminate form 0/0. then, evaluate the limit. for #8 and #9 - use the following graphs and the limit laws to evaluate each limit. 8. 9. 10. in physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by coulomb’s law: where e represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and 1/4πε0 is coulomb’s constant: 8.988 × 109 n · m2 /c2. a. use a graphing calculator to graph e(r) given that the charge of the particle is q = 10−10. b. evaluate what is the physical meaning of this quantity? is it physically relevant? why are you evaluating from the right? for #11 and #12 - determine the point(s), if any, at which each function is discontinuous. classify any discontinuity as jump, removable, infinite, or other. 11. 12. 13. decide if the function is continuous at the given point. if it is discontinuous, what type of discontinuity is it? for #14 and #15 - find the value(s) of k that makes each function continuous over the given interval. 14. 15. 16. 17. for #18 - #20 - find δ in terms of ε using the formal limit definition. 18. 19. 20. module 6—written assignment 3 1. for the following position function y = s(t) = t2 − 2t, an object is moving along a straight line, where t is in seconds and s is in meters. find: a. the simplified expression for the average velocity from t = 2 to t = 2 + h; b. the average velocity between t = 2 and t = 2 + h, where (i) h = 0.1, (ii) h = 0.01, (iii) h = 0.001, and (iv) h = 0.0001; and c. use the answer from a. to estimate the instantaneous velocity at t = 2 second. 2. for the function f (x) = x3 − 2x2 − 11x + 12, do the following. a. use a graphing calculator to graph f in an appropriate viewing window. b. use the zoom feature on the calculator (or web graphing) to approximate the two values of x = a for which mtan = f ′(a) = 0. 3. for the following exercises, use the definition of a derivative to find f ′(x). f(x) = 4x2 4. for the following exercises, use the graph of y = f (x) to sketch the graph of its derivative f ′(x). 5. suppose temperature t in degrees fahrenheit at a height x in feet above the ground is given by y = t(x). a. give a physical interpretation, with units, of t′(x). b. if we know that t′ (1000) = −0.1, explain the physical meaning. 6. find the equation of the tangent line t(x) to the graph of the given function at the indicated point. use a graphing calculator to graph the function and the tangent line. 7. assume that f (x) and g(x) are both differentiable functions for all x. find the derivative of the function h(x). 8. assume that and are both differentiable functions with values as given in the following table. use the following table to calculate the following derivative. 9. find a quadratic polynomial such that f (1) = 5, f ′ (1) = 3 and f ″(1) = −6. 10. the given function represents the position of a particle traveling along a horizontal line. a. find the velocity and acceleration functions. b. determine the time intervals when the object is slowing down or speeding up. 11. a ball is thrown downward with a speed of 8 ft/s from the top of a 64-foot-tall building. after t seconds, its height above the ground is given by s(t) = −16t2 − 8t + 64. a. determine how long it takes for the ball to hit the ground. b. determine the velocity of the ball when it hits the ground. 12. find the derivative dy/dx for 13. find the equation of the tangent line to the given function at the indicated values of x. then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct. 14. find the second derivative for y = sin x cos x 15. the amount of rainfall per month in phoenix, arizona, can be approximated by y(t) = 0.5 + 0.3cost, where t is months since january. find y′ and use a calculator to determine the intervals where the amount of rain falling is decreasing. 16. find the derivative for the function: 17. find the equation of the tangent line to the graph of the given equation at the indicated point. use a calculator or computer software to graph the function and the tangent line. 18. if the surface area of the rectangular box is 78 square feet, find dy/dx when x = 3 feet and y = 5 feet. 19. find the derivative for 20. find the equation of the tangent line to the graph of x3 − xlny + y3 = 2x + 5 at the point where x = 2. (hint: use implicit differentiation to find dy/dx.) graph both the curve and the tangent line.) module 8—written assignment 4 1. 2. you and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. you both leave from the same point, with you riding at 16 mph east and your friend riding 12 mph north. after 2π.="" 11.="" find="" a.="" the="" amplitude,="" b.="" the="" period,="" and="" c.="" the="" phase="" shift="" with="" direction="" for="" each="" function.="" 12.="" a.="" find="" the="" inverse="" function,="" and="" b.="" find="" the="" domain="" and="" range="" of="" the="" inverse="" function.="" 13.="" evaluate="" the="" function.="" give="" the="" exact="" value.="" 14.="" the="" depth="" (in="" feet)="" of="" water="" at="" a="" dock="" changes="" with="" the="" rise="" and="" fall="" of="" tides.="" it="" is="" modeled="" by="" the="" function="" where="" t="" is="" the="" number="" of="" hours="" after="" midnight.="" determine="" the="" first="" time="" after="" midnight="" when="" the="" depth="" is="" 11.75="" ft.="" 15.="" write="" the="" equation="" in="" equivalent="" exponential="" form.="" 16.="" write="" the="" equation="" in="" equivalent="" logarithmic="" form.="" for="" #17="" and="" #18="" -="" use="" properties="" of="" logarithms="" to="" write="" the="" expressions="" as="" a="" sum,="" difference,="" and/or="" product="" of="" logarithms.="" 17.="" 18.="" for="" #="" 19="" and="" #="" 20="" -="" use="" the="" change-of-base="" formula="" and="" either="" base="" 10="" or="" base="" e="" to="" evaluate="" the="" given="" expressions.="" answer="" in="" exact="" form="" and="" in="" approximate="" form,="" rounding="" to="" four="" decimal="" places.="" 19.="" 20.="" module="" 4—written="" assignment="" 2="" 1.="" points="" p(4,="" 2)="" and="" q(x,="" y)="" are="" on="" the="" graph="" of="" the="" function="" f="" (x)="." complete="" the="" following="" table="" with="" the="" appropriate="" values:="" y-coordinate="" of="" q,="" the="" point="" q(x,="" y),="" and="" the="" slope="" of="" the="" secant="" line="" passing="" through="" points="" p="" and="" q.="" round="" your="" answer="" to="" eight="" significant="" digits.="" 2.="" use="" the="" value="" in="" the="" preceding="" exercise="" to="" find="" the="" equation="" of="" the="" tangent="" line="" at="" point="" p.="" 3.="" consider="" the="" function="" f="" (x)="−x2" +="" 1="" sketch="" the="" graph="" of="" f="" over="" the="" interval="" [−1,="" 1].="" 4.="" approximate="" the="" area="" of="" the="" region="" between="" the="" x-axis="" and="" the="" graph="" of="" f="" over="" the="" interval="" [−1,="" 1]="" for="" problem="" 3.="" 5.="" use="" the="" graph="" of="" the="" function="" y="f" (x)="" shown="" here="" to="" find="" the="" values="" of="" the="" four="" limits,="" if="" possible.="" estimate="" when="" necessary.="" 6.="" use="" direct="" substitution="" to="" show="" that="" each="" limit="" leads="" to="" the="" indeterminate="" form="" 0/0.="" then,="" evaluate="" the="" limit.="" 7.="" use="" direct="" substitution="" to="" show="" that="" each="" limit="" leads="" to="" the="" indeterminate="" form="" 0/0.="" then,="" evaluate="" the="" limit.="" for="" #8="" and="" #9="" -="" use="" the="" following="" graphs="" and="" the="" limit="" laws="" to="" evaluate="" each="" limit.="" 8.="" 9.="" 10.="" in="" physics,="" the="" magnitude="" of="" an="" electric="" field="" generated="" by="" a="" point="" charge="" at="" a="" distance="" r="" in="" vacuum="" is="" governed="" by="" coulomb’s="" law:="" where="" e="" represents="" the="" magnitude="" of="" the="" electric="" field,="" q="" is="" the="" charge="" of="" the="" particle,="" r="" is="" the="" distance="" between="" the="" particle="" and="" where="" the="" strength="" of="" the="" field="" is="" measured,="" and="" 1/4πε0="" is="" coulomb’s="" constant:="" 8.988="" ×="" 109="" n="" ·="" m2="" c2.="" a.="" use="" a="" graphing="" calculator="" to="" graph="" e(r)="" given="" that="" the="" charge="" of="" the="" particle="" is="" q="10−10." b.="" evaluate="" what="" is="" the="" physical="" meaning="" of="" this="" quantity?="" is="" it="" physically="" relevant?="" why="" are="" you="" evaluating="" from="" the="" right?="" for="" #11="" and="" #12="" -="" determine="" the="" point(s),="" if="" any,="" at="" which="" each="" function="" is="" discontinuous.="" classify="" any="" discontinuity="" as="" jump,="" removable,="" infinite,="" or="" other.="" 11.="" 12.="" 13.="" decide="" if="" the="" function="" is="" continuous="" at="" the="" given="" point.="" if="" it="" is="" discontinuous,="" what="" type="" of="" discontinuity="" is="" it?="" for="" #14="" and="" #15="" -="" find="" the="" value(s)="" of="" k="" that="" makes="" each="" function="" continuous="" over="" the="" given="" interval.="" 14.="" 15.="" 16.="" 17.="" for="" #18="" -="" #20="" -="" find="" δ="" in="" terms="" of="" ε="" using="" the="" formal="" limit="" definition.="" 18.="" 19.="" 20.="" module="" 6—written="" assignment="" 3="" 1.="" for="" the="" following="" position="" function="" y="s(t)" =="" t2="" −="" 2t,="" an="" object="" is="" moving="" along="" a="" straight="" line,="" where="" t="" is="" in="" seconds="" and="" s="" is="" in="" meters.="" find:="" a.="" the="" simplified="" expression="" for="" the="" average="" velocity="" from="" t="2" to="" t="2" +="" h;="" b.="" the="" average="" velocity="" between="" t="2" and="" t="2" +="" h,="" where="" (i)="" h="0.1," (ii)="" h="0.01," (iii)="" h="0.001," and="" (iv)="" h="0.0001;" and="" c.="" use="" the="" answer="" from="" a.="" to="" estimate="" the="" instantaneous="" velocity="" at="" t="2" second.="" 2.="" for="" the="" function="" f="" (x)="x3" −="" 2x2="" −="" 11x="" +="" 12,="" do="" the="" following.="" a.="" use="" a="" graphing="" calculator="" to="" graph="" f="" in="" an="" appropriate="" viewing="" window.="" b.="" use="" the="" zoom="" feature="" on="" the="" calculator="" (or="" web="" graphing)="" to="" approximate="" the="" two="" values="" of="" x="a" for="" which="" mtan="f" ′(a)="0." 3.="" for="" the="" following="" exercises,="" use="" the="" definition="" of="" a="" derivative="" to="" find="" f="" ′(x).="" f(x)="4x2" 4.="" for="" the="" following="" exercises,="" use="" the="" graph="" of="" y="f" (x)="" to="" sketch="" the="" graph="" of="" its="" derivative="" f="" ′(x).="" 5.="" suppose="" temperature="" t="" in="" degrees="" fahrenheit="" at="" a="" height="" x="" in="" feet="" above="" the="" ground="" is="" given="" by="" y="T(x)." a.="" give="" a="" physical="" interpretation,="" with="" units,="" of="" t′(x).="" b.="" if="" we="" know="" that="" t′="" (1000)="−0.1," explain="" the="" physical="" meaning.="" 6.="" find="" the="" equation="" of="" the="" tangent="" line="" t(x)="" to="" the="" graph="" of="" the="" given="" function="" at="" the="" indicated="" point.="" use="" a="" graphing="" calculator="" to="" graph="" the="" function="" and="" the="" tangent="" line.="" 7.="" assume="" that="" f="" (x)="" and="" g(x)="" are="" both="" differentiable="" functions="" for="" all="" x.="" find="" the="" derivative="" of="" the="" function="" h(x).="" 8.="" assume="" that="" and="" are="" both="" differentiable="" functions="" with="" values="" as="" given="" in="" the="" following="" table.="" use="" the="" following="" table="" to="" calculate="" the="" following="" derivative.="" 9.="" find="" a="" quadratic="" polynomial="" such="" that="" f="" (1)="5," f="" ′="" (1)="3" and="" f="" ″(1)="−6." 10.="" the="" given="" function="" represents="" the="" position="" of="" a="" particle="" traveling="" along="" a="" horizontal="" line.="" a.="" find="" the="" velocity="" and="" acceleration="" functions.="" b.="" determine="" the="" time="" intervals="" when="" the="" object="" is="" slowing="" down="" or="" speeding="" up.="" 11.="" a="" ball="" is="" thrown="" downward="" with="" a="" speed="" of="" 8="" ft/s="" from="" the="" top="" of="" a="" 64-foot-tall="" building.="" after="" t="" seconds,="" its="" height="" above="" the="" ground="" is="" given="" by="" s(t)="−16t2" −="" 8t="" +="" 64.="" a.="" determine="" how="" long="" it="" takes="" for="" the="" ball="" to="" hit="" the="" ground.="" b.="" determine="" the="" velocity="" of="" the="" ball="" when="" it="" hits="" the="" ground.="" 12.="" find="" the="" derivative="" dy/dx="" for="" 13.="" find="" the="" equation="" of="" the="" tangent="" line="" to="" the="" given="" function="" at="" the="" indicated="" values="" of="" x.="" then="" use="" a="" calculator="" to="" graph="" both="" the="" function="" and="" the="" tangent="" line="" to="" ensure="" the="" equation="" for="" the="" tangent="" line="" is="" correct.="" 14.="" find="" the="" second="" derivative="" for="" y="sin" x="" cos="" x="" 15.="" the="" amount="" of="" rainfall="" per="" month="" in="" phoenix,="" arizona,="" can="" be="" approximated="" by="" y(t)="0.5" +="" 0.3cost,="" where="" t="" is="" months="" since="" january.="" find="" y′="" and="" use="" a="" calculator="" to="" determine="" the="" intervals="" where="" the="" amount="" of="" rain="" falling="" is="" decreasing.="" 16.="" find="" the="" derivative="" for="" the="" function:="" 17.="" find="" the="" equation="" of="" the="" tangent="" line="" to="" the="" graph="" of="" the="" given="" equation="" at="" the="" indicated="" point.="" use="" a="" calculator="" or="" computer="" software="" to="" graph="" the="" function="" and="" the="" tangent="" line.="" 18.="" if="" the="" surface="" area="" of="" the="" rectangular="" box="" is="" 78="" square="" feet,="" find="" dy/dx="" when="" x="3" feet="" and="" y="5" feet.="" 19.="" find="" the="" derivative="" for="" 20.="" find="" the="" equation="" of="" the="" tangent="" line="" to="" the="" graph="" of="" x3="" −="" xlny="" +="" y3="2x" +="" 5="" at="" the="" point="" where="" x="2." (hint:="" use="" implicit="" differentiation="" to="" find="" dy/dx.)="" graph="" both="" the="" curve="" and="" the="" tangent="" line.)="" module="" 8—written="" assignment="" 4="" 1.="" 2.="" you="" and="" a="" friend="" are="" riding="" your="" bikes="" to="" a="" restaurant="" that="" you="" think="" is="" east;="" your="" friend="" thinks="" the="" restaurant="" is="" north.="" you="" both="" leave="" from="" the="" same="" point,="" with="" you="" riding="" at="" 16="" mph="" east="" and="" your="" friend="" riding="" 12="" mph="" north.="">