Spring Homework Part B Packet. MAT Calculus I

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1 Class: MAT Spring 2015 Homework Part B Packet What you will find in this packet: Assignment Directions Class Assignments o Reminders to do your Part A problems ( o All of your Part B assignments Dr. Kristyanna Erickson Professor of Mathematics Cecil College KErickson@cecil.edu Web Site: Facebook:

2 Homework Part B Packet Trig Identities FUNDAMENTAL TRIGONOMETRIC IDENTITIES Dr. Kristyanna Erickson 2 Reciprocal Identities 1 1 sin x= csc x= csc x sin x 1 1 cos x= sec x= sec x cos x 1 1 tan x= cot x= cot x tan x Quotient Identities sin x cos x tan x= cot x= cos x sin x Pythagorean Identities x+ x= 2 2 sin cos 1 Even-Odd Identities 1+ cot x= csc 2 2 tan x+ 1 = sec 2 2 sin( x) = sin x cos( x) = cos( x) tan( x) = tan( x) csc( x) = csc x sec( x) = sec( x) cot( x) = cot( x) x x OTHER TRIGONOMETRIC IDENTITIES SUM AND DIFFERENCE FORMULAS ( ) ( ) ( ) ( ) sin α + β = sinαcos β + cosαsin β sin α β = sinαcos β cosαsin β cos α + β = cosαcos β sinαsin β cos α β = cosαcos β + sinαsin β tan tan tanα + tan β + = 1 tanα tan β ( α β) tanα tan β = 1 + tanα tan β ( α β) Power-Reducing Formulas 2 1 cos 2θ sin θ = cos 2θ cos θ = cos 2θ tan θ = 1 + cos 2θ Half-Angle Formulas α 1 cosα sin = ± 2 2 α 1+ cosα cos = ± 2 2 α 1 cosα 1 cosα sinα tan =± = = cos α sin α 1 + cos α Double-Angle Formulas sin 2θ = 2sinθcosθ 2 2 θ = θ θ cos 2 cos sin = θ 2 2cos 1 2 = 1 2sin θ 2 tanθ tan 2θ = 2 1 tan θ OBLIQUE TRIANGLES LAW OF SINES a = b = c sin A sin B sin C b C a LAW OF COSINES = + 2 cos cos b + c a = 2bc = + 2 cos cos a + c b = 2ac = + 2 cos cos a + b c = 2ab a b c bc A A b a c ac B B c a b ab C C A c B

3 Homework Part B Packet Dr. Kristyanna Erickson 3 Assignment Directions Part A problems These problems are graded by the computer and are posted immediately on Web Assign. The computer will stop accepting Part A work at 11:55 PM on the due date. Incorrect problems may be redone up to six times for full credit until the due date. Technological problems arise, so don t wait until 11:00 p.m. the night assignments are due to begin. Technological issues are not an acceptable excuse for late/missed assignments. IMORTANT: Do not wait until the last minute to submit an assignment on WebAssign. Assignment cutoff times are determined by the clock of the WebAssign server, not by the clock on your computer. Every effort is made to ensure that these server clocks are accurate. If the assignment cutoff time is 11:55 P.M., you cannot submit the assignment after 11:55 P.M. according to the WebAssign server, regardless of the time displayed on your computer. Part B problems One of the main purposes of Part B problems is to see if in solving problems you can write out all of the steps correctly using proper mathematical notation and to communicate your solutions in a wellorganized and readable fashion. Sloppy work, poor organization, and poor notation may result in loss of credit even when the answer is correct. You may discuss problems with other students in this class, but your final work must be your own. Neatness counts - Sloppy work, poor organization, and poor notation may result in loss of credit even when the answer is correct. Must use pencil. Full, proper notation must be used at all times or you will not receive full credit. You must show your work. You may ask me for help with these problems. All work must be submitted on time, or early. Your lowest 2 Part B Homework assignments will be dropped. This includes excused and unexcused absences and lowest grades. If you use a calculator or other technology for graphing you must sketch a proper graph on your assignment. Likewise if you use a calculator or other technology to calculate an answer you should show what you input into the device. Decimal approximations should not be used unless they are specifically asked for. For example if your calculator gives an answer as , which is an approximation for the fraction 1/6, you should write the answer as 1/6 or you may lose credit. IMPORTANT REMINDERS - Remember, late work will not be accepted, so if you are not in class on the due date for an assignment, be sure the problems are handed in prior to the due date. You may hand in work to me during office hours, to the Math Lab (make sure you sign the student drop off sheet and write the same date and time on your paper), or by fax/ . Be sure it is handed in before the due date. If your work is not handed in by the due date or these directions are not followed, then the assignment will not be accepted and a zero will be assigned. The two lowest assignment grades will be dropped at the end of the semester.

4 Homework Part B Packet Dr. Kristyanna Erickson 4 Helpful Homework Hints The symbol f(x) is read as f of x and it replaces the letter y in an equation. If you prefer using the letter y instead of f(x) just rewrite the equation using y in your work and then replace the letter y with f(x) at the end. To graph a radical equation, use the key followed by parenthesis containing what is inside the radical sign. Example: 2x 3x 5 is entered as 2 x (3x 5). Since the square root means the one-half power, you could also have the following 2x( 3x 5) 1/2 entered on your calculator as 2 x(3x 5) ^ (1/ 2). To enter the absolute value into an equation, use the abs( key on the calculator or go to the catalog on the calculator and find abs(. Remember, the quantity in absolute value must be in parenthesis on the calculator. Whenever you are asked to sketch a function using the graphing calculator, you are expected to draw a proper sketch of the graph on your paper. A proper sketch has the x- and y-axis labeled, the scale labeled, asymptotes labeled and dotted, x and y intercepts if any labeled, maximums and minimums labeled if any, starting points on graphs if any, and arrows on the function showing the function continues. Remember that when you are inputting an algebraic fraction into a calculator, the top and bottom must be enclosed in parenthesis. All answers should be given as exact answers. Sometimes a problem asks for exact and approximate answers. When rounding, please round to 4 decimal places unless stated otherwise. Here is an example. for x stating both the exact and approximate answers. Solve the equation 3x = 10 3x = x = 3 x Strategy for solving word problems: 1. Restate the problem in your own words and draw a diagram 2. Define the unknown(s) that you are trying to find 3. Write a formula or equation 4. Solve the equation Make sure your answer answers the question and is reasonable.

5 HW 1B HW 1 B Review 1.1. Draw the graph of the piecewise function. 1 if x < 1 f( x) = 3x+ 2 if x < 1 7 2x if x 1

6 HW 1B 1.2. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft. express the area A of the window as function of the width x of the window.

7 HW 1B 1.3. Express the function F( x) = 1 x+ x as a composition of three functions.

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9 HW 2B HW 2 B Section The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2sinπt+ 3cosπt, where t is measured in seconds. a. Find the average velocity during each time period: i. [1, 2] ii. [1, 1.1] iii. [1,1.01] iv. [1,1.001] b. Estimate the instantaneous velocity of the particle when t = 1 to two decimal places.

10 HW 2B Section For the function g whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. a. lim gt ( ) t 0 e. lim gt ( ) + t 2 b. lim gt ( ) + t 0 f. lim gt ( ) t 2 c. lim gt ( ) t 0 g. g (2) d. lim gt ( ) t 2 lim gt ( ) h. t 4

11 HW 3B Section Algebraically prove that + x 0 sin( π x) HW 3 B lim xe = 0.

12 HW 3B 3.2. Evaluate the limit, if it exists. a. 2 x 4x lim x 4 2 x 3x 4 b. lim x 4 2 x x + 4

13 HW 3B Section The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is GMr if r < R 3 Fr () = R GM if r R 2 r where M is the mass of the Earth, R is the radius, and G is the gravitational constant. Is F a continuous function of r? Explain using calculus terminology and notation.

14 HW 3B Section Sketch the graph of an example of a function f that satisfies all of the given conditions. f (0) = 3, lim f( x) = 4, lim f( x) = 2, lim f( x) =, + x 0 x 0 lim f( x) =, lim f( x) =, lim f( x) = 3 + x 4 x 4 x x

15 HW 4B HW 4B Section A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. a. When is the particle moving to the right? b. When is the particle moving to the left? c. When is the particle standing still? d. Draw a graph of the velocity function.

16 HW 4B Section The figure shows the graph of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

17 HW 4B Section The equation y" y' 2y x function y and its derivatives 2 + = is called a differential equation because it involves an unknown 2 y = Ax + Bx + C satisfies this equation. y ' and y ". Find constants AB,, and, C such that the function

18 HW 4B Section 3.2 x 4.4. The curve y = is called a serpentine x a. Find an equation of the tangent line to this curve at the point ( 3,0.3 ) b. Find an equation of the normal line to this curve at the point ( 3,0.3 )

19 HW 5B HW 5B Section A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of π the ladder slides away from the wall, how fast does x change with respect to θ when θ =? 3

20 HW 5B Section Find the derivatives of the following functions showing your work and fully simplifying your answers. Using the following rules as necessary: Product, Quotient and Chain rules. You may not use logarithmic differentiation. a. f ( x) = ( ax + b) 3 ( cx + d ) 2 4 b. sin f( x) = cos ( ax) ( bx)

21 HW 5B 3 5 c. f( x) = ( x+ 3) ( x 8) d. f( x) = ( 8x 3) ( 6x + 7) 5 12

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23 HW 6B HW 6B Section Show that the sum of the x and y intercepts of any tangent line to the curve x + y = c is equal to c.

24 HW 6B Section Differentiate f and find the domain of f f( x) = ln ln ln x y = sin x 6.3. Use logarithmic differentiation to find the derivative of ( ) ln x

25 HW 7B Section Prove the identity cosh x+ sinh x= e x HW 7B

26 HW 7B Section A particle moves with position function s = t 4t 20t + 20t t 0 a. At what time does the particle have a velocity of 20 m/s? b. At what time is the acceleration 0? What is the significance of this value of t.

27 HW 7B Section In a murder investigation, the temperature of the corpse was 32.5 C at 1:30 PM and 30.3 C an hour later. Normal body temperature is 37.0 C and the temperature of the surroundings was 20.0 C. When did the murder take place?

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29 HW 8B HW 8B Section When air expands adiabatically (without gaining or losing heat), its pressure P and volume V 1.4 are related by the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 400cm 3 and the pressure is 80kPa and is decreasing at a rate of 10kPa/min. At what rate is the volume increasing at this instant?

30 HW 8B Section One side of a right triangle is known to be 20 cm long and the opposite angle is measured at 30, with a possible error of ±1. a. Use differentials to estimate the error in computing the length of the hypotenuse. b. What is the percentage error?

31 HW 9B Section 4.1 HW 9B 9.1. Find the absolute maximum and absolute minimum values of f( t) = 2cost+ sin 2t on the π interval 0, 2.

32 HW 9B Section At 2:00PM a car s speedometer reads 30mi/h. At 2:10PM it reads 50mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120mi/h 2.

33 HW 9B Section f( x) = 1+ 2 x x a. Find each of the following using calculus. Show all your work. i. Any vertical asymptotes ii. Any horizontal asymptotes iii. When it is increasing iv. When it is decreasing

34 HW 9B v. Any local maximum values vi. Any local minimum values vii. When it is concave up viii. When it is concave down ix. Any inflection points b. Sketch a proper graph of the function

35 HW10B HW 10B Section 4.4 ln x Use l Hospital s Rule to prove that lim = 0 for any number p > 0. This shows that the x p x logarithmic function approaches more slowly than any power of x.

36 HW10B Section A cylindrical can without a top is made to contain V cm 3 of liquid. Find the dimensions that will minimize the cost of metal to make the can.

37 HW11B HW 11B Section A particle is moving with the given data. Find the position of the particle. at = t t+ s = s = 2 ( ) 4 6, (0) 0, (1) 20

38 HW11B Section Determine a region whose area is equal to the given limit. Do not evaluate the limit. n 2 2i lim 5 + n i 1 n n = 10

39 HW12B HW 12B Section Use the definition of a definite integral from Section 5.2 to evaluate the integral ( 4 + 2) x x dx. Remember this is the limit of a sum approach.

40 HW12B Section Evaluate the integrals (you may use the Fundamental Theorem of Calculus, Part 2, but you may not use substitution.) 2 a. ( + ) y dy b. 1 1 e u+ 1 du

41 HW12B c v + 3v 4 dv v

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43 HW13B Section Evaluate ee xx sinh xx + cosh xx dddd HW 13B

44 HW13B Section Evaluate the indefinite integrals sin x a. dx 2 1+ cos x x b. dx 4 1+ x