MA139 Bentley University Review Sheet #1 2024 PDF

MA139 Bentley University CICCARELLI Review Sheet #1 For Exam #1, you are responsible for all of the material covered in class (even if you were not th...

MA139 Bentley University CICCARELLI Review Sheet #1 For Exam #1, you are responsible for all of the material covered in class (even if you were not there!), and all of the homework since the beginning of the semester. If you did not do certain homework problems, I suggest you do them. If you missed any classes, get notes from someone in the class. ___________________________________________________________________________________________________________________________ You will need a scientific (or graphing) calculator for the test, so BRING ONE. Partial credit will be given on the exam, BUT ONLY IF YOU SHOW YOUR WORK! NO homework from Sections 4.9 – 5.5 will be accepted after the test, so catch up if you need to. ___________________________________________________________________________________________________________________________ By far, the best way to study for a math test is to roll up your sleeves and do lots & lots of problems. It doesn’t matter if you wear gloves. These should get you started. You might also want to try some problems from Section 4.9 and the Chapter 5 Review, as well as the Calculus Resources Site. If you can do the problems on this Review Sheet (on your own!), you should be fine for the exam. So get at it, post-haste! Your Fearless Leader, Pete MA139 Practice Problems for Test #1 CICCARELLI Sections 4.9 – 5.5 Here are some extra practice problems related to the topics we’ve studied in class. These are not due; however, problems like these (possibly exactly like these) may be on the quiz &/or test. Feel free to work together, bouncing ideas off of each other, but each person should be able to write up their own solutions, as you’ll need to on a quiz or test. 1.) Let 𝑓(𝑥) = 𝑥 2 + 𝑥 − 6 on the interval [1, 4]. 4 a) Estimate the signed area ∫1 (𝑥 2 + 𝑥 − 6) 𝑑𝑥 (by hand, not Winplot) using 𝑛 = 6 equal subintervals and: i) 𝑅6 ii) 𝐿6 iii) 𝑀6 b) Find the exact signed area by using 𝑅𝑛 and equal subintervals and letting 𝑛 → ∞. Then check your answer using the FTC. 2.) (Riemann Sums ↔ Definite Integrals) a) Write the definite integral as a limit of a Riemann Sum. DO NOT SIMPLIFY OR TRY TO EVALUATE! 8 4𝑥 2 2𝜋 i) ∫3 𝑑𝑥 ii) ∫0 𝑒 −𝑥 sin(3𝑥) 𝑑𝑥 √1+𝑥 3 b) Write the Riemann Sum limit as a definite integral: 3𝑖 2 3 i) lim ∑𝑛𝑖=1 ln [(2 + ) + 1] ( ) 𝑛→∞ 𝑛 𝑛 𝜋 𝜋𝑖 5 sec( + ) 𝜋 ii) lim ∑𝑛𝑖=1 4 12𝑛 𝜋 𝜋𝑖 (12𝑛) 𝑛→∞ ( + )+1 4 12𝑛 3.) A custom-made designer swimming pool was measured at 3 ft. intervals, as shown: 𝑥 ft. from left end 0 3 6 9 12 15 18 21 24 𝑦 ft. wide at pt. 𝑥 0 18.7 21.1 20.2 18.1 16.8 15 14.7 0 a) Use 𝑅8 to estimate the surface area of the pool. b) If the pool is 4 ft. deep everywhere, what volume of water does it hold? (Use your answer to part a to estimate this.)  𝑏 4.) Give the limit definition of the definite integral ∫𝑎 𝑓(𝑥)𝑑𝑥, and explain in your own words what each piece of the formula means. 5.) Find the following signed areas by sketching the graph and using geometry: 3 a) ∫−3 −√9 − 𝑥 2 𝑑𝑥 4 b) ∫−4 3 − |𝑥| 𝑑𝑥 −𝑥 − 2 for 𝑥 in (−∞, 0) 9 1 c) ∫−2 𝑓(𝑥)𝑑𝑥, where 𝑓(𝑥) = { 𝑥 − 2 for 𝑥 in [0, 6] } 2 3 for 𝑥 in (6, ∞) 6.) Use the properties of the definite integral to… 3 3 7 a) Find ∫7 𝑓(𝑥)𝑑𝑥, given that ∫1 𝑓(𝑥)𝑑𝑥 = 4 and ∫1 𝑓(𝑥)𝑑𝑥 = 10. 75 b) Show that ∫74 (1 + cos 𝑥)𝑑𝑥 ≥ 0. 2 1 c) Estimate (find upper & lower bounds for) ∫0 1+𝑥 2 𝑑𝑥. 7.) Goin’ Old School (with new notation): Archimedes & the Method of Exhaustion Let’s find the area of a circle! (Because you have nothing more pressing to do right now…) Let 𝑃𝑛 be a regular 𝑛-sided polygon, inscribed in a circle of radius 𝑟, where 𝑃𝑛 has sides of length 𝑠 and apothem of length 𝑎. “What’s an “apothem”, Pete?” Heck, here’s a picture: 2𝜋 Note that each “sector” of 𝑃𝑛 has angle 𝑛. a) Show that a right triangle comprising half of one of these sectors (see right picture above) has area 1 𝜋 𝜋 1 𝐴∆ = 2 𝑟 2 sin (𝑛) cos (𝑛 ) by using “SOHCAHTOA” & the fact that 𝐴∆ = 2 𝑏ℎ. 𝜋 𝜋 b) Show the area of 𝑃𝑛 is 𝐴𝑃𝑛 = 𝑛𝑟 2 sin (𝑛) cos (𝑛) by adding up all of the “half”-triangles comprising 𝑃𝑛. c) Now let the # of sides 𝑛 → ∞, so that we “exhaust” all of the area inside the circle. That is, show 𝜋 lim 𝐴𝑃𝑛 = 𝜋𝑟 2. (HINT: In the formula in part b, multiply the top & bottom by and use the fact that 𝑛→∞ 𝑛 sin 𝜃 lim = 1, or L’Hospital’s Rule.) Now who’s exhausted…? 𝜃→0 𝜃  0 for 𝑥 = 0 8.) Show that 𝑓(𝑥) = { [0, for 𝑥 in (0, 1]} is not integrable on 1] by showing that the limit definition of the 1 𝑥 definite integral doesn’t exist (in fact, goes to +∞) for the following partition of [0, 1]: 1−0 1 1 Use equal subintervals of width ∆𝑥 = = , and in the first subinterval [𝑥0 , 𝑥1 ] = [0, ] choose the first 𝑛 𝑛 𝑛 1 1 sample pt. to be 𝑥1∗ = 𝑛2 , which is clearly in the interval [0, 𝑛]. You can pick the other sample pts. 𝑥𝑖∗ in the other subintervals to be whatever you want – it won’t matter. Show that just because of the first term in the Riemann Sum alone, we’ll have lim ∑𝑛𝑖=1 𝑓(𝑥𝑖∗ ) ∙ ∆𝑥 = +∞, so 𝑓 is not integrable on [0, 1]. 𝑛→∞ 9.) State the 1st & 2nd FTC, and explain what they mean in your own words. 10.) Use the 1st FTC to find the derivatives of the following functions: 𝑥 sin 𝑡 a) 𝐹(𝑥) = ∫2 𝑑𝑡 𝑡 3 b) 𝐺(𝑥) = ∫𝑥 √1 + 𝑡 4 𝑑𝑡 𝑥5 c) 𝑦 = ∫1 𝑒 𝑡 cos 𝑡 𝑑𝑡 ln 𝑥 d) 𝐻(𝑥) = ∫5𝑥 tan−1 (𝑡 2 ) 𝑑𝑡 11.) FILL IN THE BLANKS: “Know Thy Antiderivative Rules” ∫ 𝑥 𝑛 𝑑𝑥 = ____________, except when 𝑛 = ___ ∫ sec 2 𝑥 𝑑𝑥 = ______________________ 1 ∫ 𝑑𝑥 = ___________________ ∫ csc 2 𝑥 𝑑𝑥 = ___________________ 𝑥 ∫ 𝑒 𝑥 𝑑𝑥 = ___________________ ∫ sec 𝑥 tan 𝑥 𝑑𝑥 = ___________________ ∫ sin 𝑥 𝑑𝑥 = ___________________ ∫ csc 𝑥 cot 𝑥 𝑑𝑥 = ___________________ 1 ∫ cos 𝑥 𝑑𝑥 = ___________________ ∫ 𝑑𝑥 = ___________________ √𝑎2 − 𝑥 2 1 ∫ tan 𝑥 𝑑𝑥 = ___________________ ∫ 2 𝑑𝑥 = ___________________ 𝑎 + 𝑥2 1 ∫ cot 𝑑𝑥 = ___________________ ∫ 𝑑𝑥 = ___________________ 𝑥 ∙ √𝑥 2 − 𝑎2 ∫ sec 𝑥 𝑑𝑥 = ___________________ ∫ 𝑎 𝑥 𝑑𝑥 = ___________________ ∫ csc 𝑥 𝑑𝑥 = ___________________ ∫ 𝑒 𝑘𝑥 𝑑𝑥 = ___________________ 12.) Evaluate the following integrals. Be sure to distinguish between indefinite integrals (families of antiderivatives) and definite integrals (signed areas under curves)! 4 3 a) ∫1 (6𝑥 3 − 2𝑥 + − 5) 𝑑𝑥 √𝑥 4 2 b) ∫ (1.5𝑒 𝑥 + 𝑥 − 𝑥 4 + 3 sec 2 𝑥) 𝑑𝑥 𝜋⁄ c) ∫0 4(sec 2 𝑥 − tan2 𝑥) 𝑑𝑥 𝑥 2 +𝑥−1 d) ∫ 𝑑𝑥 √𝑥 2 1 e) ∫0 (𝑥−1)2 𝑑𝑥 3 𝑥+1 f) ∫2 𝑥 2 −1 𝑑𝑥 𝑥+1 g) ∫ 𝑥 2 +1 𝑑𝑥 sin 𝑥−cos 𝑥 h) ∫ 𝑑𝑥 sin 𝑥+cos 𝑥 0 3 +1 i) ∫−1 𝑥 2 𝑒 2𝑥 𝑑𝑥 4 √𝑥 j) ∫ 𝑑𝑥 √𝑥 k) ∫ tan2 𝑥 𝑑𝑥 2𝑥 3 −2 l) ∫ (𝑥 4 −4𝑥+9)6 𝑑𝑥 2 m) ∫−1(𝑥 2 − |𝑥|) 𝑑𝑥 2 2𝑥𝑒 𝑥 n) ∫ 2 2 𝑑𝑥 (𝑒 𝑥 +1) 5 o) ∫5 sin 𝑥 ∙ ln(𝑥 2 + 1) 𝑑𝑥 4 𝑥+1 for 𝑥 < 1 p) ∫0 𝑓(𝑥) 𝑑𝑥, where 𝑓(𝑥) = { 2 } 𝑥 − 4𝑥 + 4 for 𝑥 ≥ 1 4 q) ∫3 𝑥 ∙ √𝑥 − 3 𝑑𝑥 r) ∫ sin 𝑥 ∙ cos 𝑥 𝑑𝑥 (Try it 2 different ways!) s) ∫(sec 𝑥 + tan 𝑥)𝑑𝑥 5 t) ∫ 𝑑𝑥 √9−4𝑥 2  𝜋⁄ cos 𝑥 u) ∫0 2 𝑑𝑥 (How appropriate to end on “𝑢”! ) 1+sin2 𝑥 2 sin 𝑥 13.) Without integrating, explain why ∫−2 1+𝑥2 𝑑𝑥 = 0. (Graph it if you must – you shouldn’t need to!) 1 14.) Find the specific antiderivative 𝐹(𝑥) of 𝑓(𝑥) = − 4 where 𝐹(1) = 0. √𝑥 100 15.) A company has marginal revenue 𝑅 ′ (𝑥) = 3 $/unit and marginal cost 𝐶 ′ (𝑥) =.4𝑥 $/unit, when 𝑥 units are √𝑥 produced and sold. If the profit for 27 units is $520, find the profit function 𝑃(𝑥) and the profit for 64 units. (HINT: 𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥), so 𝑃′ (𝑥) = 𝑤ℎ𝑎𝑡?) 16.) An object moves on a line with acceleration given by 𝑎(𝑡) = 𝑠 ′′ (𝑡) = 3𝑡 + 2 ft./sec2. If its initial velocity is 2 ft./sec and it starts from a fixed “origin” (𝑠(0) = 0), find its position after 4 seconds. 17.) A particle moves along a line with acceleration 𝑎(𝑡) = 𝑠 ′′ (𝑡) = 6𝑡 − 3 ft./sec2. If it initially starts from rest: a) Find its velocity at time 𝑡 seconds. b) Find its total distance traveled during the first 2 seconds. 18.) An initial investment of $1000 grows at a rate of 𝑃′ (𝑡) = 400𝑒.05𝑡 $/yr after 𝑡 years. Find the value of the investment after 10 years by doing: a) An indefinite integral and solving for the constant of integration “𝐶” using an initial condition, and b) A definite integral and using the “Net Change Theorem” (aka the 2nd FTC). 19.) 120 a) If oil leaks from a tank at a rate of 𝑟(𝑡) gal/min at time 𝑡 minutes, what does ∫0 𝑟(𝑡) 𝑑𝑡 represent? b) A honeybee population starts with 100 bees (at time 𝑡 = 0 weeks) and increases at a rate of 𝑛′ (𝑡) 15 bees/week. What does 100 + ∫0 𝑛′ (𝑡)𝑑𝑡 represent? 4 20.) If 𝑓(1) = 12, 𝑓 ′ is continuous and ∫1 𝑓 ′ (𝑥)𝑑𝑥 = 17, find 𝑓(4). 𝑑 ℎ(𝑥) 21.) If 𝑓 is continuous and 𝑔 & ℎ are differentiable, show [∫ 𝑓(𝑡)𝑑𝑡] = 𝑓(ℎ(𝑥)) ∙ ℎ′ (𝑥) − 𝑓(𝑔(𝑥)) ∙ 𝑔′ (𝑥). 𝑑𝑥 𝑔(𝑥) (HINT: Use the 1st FTC & the Chain Rule!) 22.) Evaluate the following: 1 𝑑 −1 a) ∫0 𝑑𝑥 [𝑒 tan 𝑥 ]𝑑𝑥 𝑑 1 −1 b) [∫ 𝑒 tan 𝑥 𝑑𝑥 0 𝑑𝑥] 𝑑 𝑥 −1 c) [∫ 𝑒 tan 𝑡 𝑑𝑥 0 𝑑𝑡] 23.) (True/False With Reasons) Determine if the statement is true or false; if true, explain why, and if false, give a counterexample to show it. a) If 𝑓 is integrable on [𝑎, 𝑏], then 𝑓 is continuous on [𝑎, 𝑏]. b) If 𝑓 is continuous on [𝑎, 𝑏], then 𝑓 is integrable on [𝑎, 𝑏]. 𝑏 𝑏 c) If 𝑓 is continuous on [𝑎, 𝑏], then ∫𝑎 𝑥 ∙ 𝑓(𝑥) 𝑑𝑥 = 𝑥 ∙ ∫𝑎 𝑓(𝑥) 𝑑𝑥. 𝑏 𝑏 d) ∫𝑎 𝑓(𝑥) 𝑑𝑥 = ∫𝑎 𝑓(𝑧) 𝑑𝑧 3 e) If 𝑓 ′ is continuous on [1, 3], then ∫1 𝑓 ′ (𝑣) 𝑑𝑣 = 𝑓(3) − 𝑓(1). 𝑏 f) If 𝑣(𝑡) is the velocity at time 𝑡 of a particle moving on a line, then ∫𝑎 𝑣(𝑡) 𝑑𝑡 gives the distance traveled during time 𝑎 ≤ 𝑡 ≤ 𝑏. 3 2 5 2 3 2 g) ∫0 𝑒 𝑥 𝑑𝑥 = ∫0 𝑒 𝑥 𝑑𝑥 + ∫5 𝑒 𝑥 𝑑𝑥 1 h) If 𝑓 has a discontinuity at 𝑥 = 0, then ∫−1 𝑓(𝑥) 𝑑𝑥 does not exist.