Math 1220: Calculus 1 Last Homework Due Monday, April 27 Course Summary 1. Basic Theory of Calculus • know the meaning of the slope of a curve and understand the difficulty in precisely defining the tangent line to a curve • be able to sketch slope and area graphs for a given curve • how to recover a sketch of f from given sketches or sign data for f 0 (and f 00 ) • the relationship between slope and area: what the Fundamental Theorem of Calculus is, and why it is true. 2. Computing Derivatives and Antiderivatives • know the derivatives of basic functions such as: powers of x; trigonometric functions; exponential functions; logarithmic functions; inverse trigonometric functions • be able to apply the basic rules to compute derivatives of sums, products and compositions of the basic functions • be able to ‘reason backwards’ and use your knowledge of derivatives to find antiderivatives 3. Evaluating Definite Integrals • know how to use your power to find antiderivatives to evaluate definite integrals • use the geometric meaning of definite integrals to evaluate or estimate their values 4. Linear Approximation • be able to apply the linear approximation technique; this includes being able to make a good choice of reference point • be able to give a rock-solid, clearly justified upper bound for the error of your approximation 5. Using Calculus to Solve Actual Problems (a) Related Rates Problems • be able to take a problem described in English and convert it to a problem of calculus; this includes providing enough explanation that the conversion can be understood and verified by an independent reader; it also includes being able to take the situation described in the problem and extract one or more equations from it • be able to solve the resulting calculus problem (b) Optimization Problems • be able to take a problem described in English and convert it to a problem of calculus; this includes providing enough explanation that the conversion can be understood and verified by an independent reader; it also includes being able to take the situation described in the problem and extract one or more equations from it • be able to solve the resulting calculus problem (c) Integration • integration allows you to solve difficult problems by cutting them into tiny easily solved pieces, and then adding the results; be able to apply this technique to a convert (with explanation) difficult problems into integral problems. • be able to solve the resulting calculus problem (if the finding the antiderivative is not beyond what we’ve learned) Plan 1. Each day, come to class with two problems (they don’t need to be from the same section) already worked on.1 I’ll mark down that you did this (and which problems you worked on) as long as there is evidence2 that you actually spent time thinking about the problems; you’ll get full homework credit for having done this. 2. At the end of class, I’ll have you sign out showing that you worked on problems in class. You’ll get homework credit for that, too. 3. On Monday, April 27, the first day of Finals Week, you can turn in as many finished solutions as you like. If you want copies for studying, then make copies for studying; you won’t get the ones you turn in to me back in time. Finished solutions will replace earlier homework grades with 4s if they are perfect. Perfect means: • • • • One problem per piece of lined paper. The problem is neatly written on the top of paper. The solution is clearly labeled and neatly written. Your solution only attempts one substantial step (or a few very simple steps) at a time, and there is a blank line between each step and the next. • Where appropriate, you have provided justification for your steps. (i.e., don’t just write down ‘ xy = z2 ’, say instead ‘because the big triangle and the little triangle are similar, xy = z2 ’.) • Any diagrams you use are clearly labeled, neatly drawn and their connection to the problem is clearly explained. • Your solution is correct and easy to understand. At any time in the next two weeks, I’ll be happy to look at your work problem and tell you if it is acceptable, and what you’d have to do to make it acceptable. I’m happy to give hints and suggestions on finding your solutions, too. 1 This work does not need to be ‘perfect’ in the sense I’ve described in part (3). Substantial progress on a solution is one kind of evidence; a well thought out question clearly explaining what is keeping you from making progress on the problem is another kind of evidence. 2 1 1.1 Basic Theory of Calculus Theory Problem 1 (a) Explain carefully but efficiently what the Fundamental Theorem of Calculus says. (b) Explain how you can use the Fundamental Theorem of Calculus to evaluate a definite integral. Problem 2 (a) Briefly describe some of the problems that arise when you attempt to give unambiguous instructions on how to draw a tangent line to a curve. (b) How did we end up defining tangent lines? (c) Explain how to get from our definition of tangent line to the limit formula f (x) − f (t) . f 0 (x) = lim t→x x−t 1.2 Practice Problem 3 (a) Here is the graph of f . Sketch a slope graph and an area graph for f . Make sure to mark all the important points on your graphs. (b) Here are the graphs of g 0 and g 00 First, decide which graph is which. Then use the graphs to sketch the graph of g . (You can do this by extracting the sign data from the graphs. If you are more ambitious, you can incorporate even more detail.) Make sure to mark all the important points on your graph. Problem 4 Let f (x) = x5 − 5x3 + x. (a) Use your calculus superpowers to sketch the graph f for all real numbers. (b) Find the maximum and minimum values of f (x) for x between − 21 and 1. 2 Computing Derivatives and Antiderivatives Problem 5 2 (a) Find the derivative of h(x) = ex cos( x12 ). (b) Find the derivative of i(x) = sin(cos(sin2 (cos3 (x4 )))). Problem 6 Let n be a positive integer. (a) Find a formula (probably involving n) for the derivative of the function k(x) = sinn (x) (which means (sin(x))n ). Rπ (b) Find a formula (probably involving n) for 02 sinn (x) cos(x) dx. 3 Evaluating Definite Integrals Problem 7 R1 √ 4 − x2 dx. R2 1 (b) Explain why 1 x4 +1 dx can be no greater than 2. Can you find a lower bound for this integral? (c) Consider the two integrals Z 2b Z b x f (x) dx and f dx. 2 2a a Draw a representative picture of regions whose area they might compute. By analyzing these regions, determine how these integrals are related: that is, suppose I tell you the first integral is, say, 49; then what can you deduce about the second integral? (a) Evaluate −1 Problem 8 (a) Evaluate Rπ (b) Evaluate R1 0 sin(x) dx. 1 0 x2 +1 dx. 4 Linear Approximation Problem 9 (a) Use a linear approximation to estimate √ 4 2. (b) Use your answer to (a) as the reference point for a second linear √ approximation and get, hopefully, an even better estimate of 4 2. (c) Find an upper bound for the error of your approximation in (b). Problem 10 (a) Find the linear function ` whose graph is the tangent line to the graph of f (x) = sin(x) at reference point a = π6 . √ √ (b) You know that sin( π4 ) = 22 , so `( π4 ) ≈ 22 . Use this approximation to derive an estimate for the number π. (c) Can you find an upper bound on the error of your approximation? 5 Using Calculus to Solve Actual Problems 5.1 Related Rates Problems Problem 11 You are a calculus instructor, trying to think of an interesting related rates problem for your students. You start by imagining a light rising up a pole while a person walks away, then you start assigning numbers: at the special time in question • the person (who is 5 feet tall) is 10 feet horizontally from the pole and walking at 3 feet per second • the light is 10 feet off the ground and moving up at ...? “This is boring,” you think, “It’s the same old problem!” But then you have an idea: if you carefully choose the motion of the light, then the length shadow of the person won’t change at all! Cackling diabolically, you set about determining the required velocity for the light. What is it? Problem 12 A big conical tank, 12 feet tall with a top radius of 4 feet is full to the top of maple syrup to dump onto the world’s biggest pancake, which is being maneuvered into place by a gang of four flatbead tractor trailors. At the chosen moment, after a suspenseful countdown of course, you pull the chain that opens the nozzle at the bottom and the syrup begins to flow. Some time later, when the level of the tank is down to 8 feet, you observe that the tank is emptying at a rate of about 1 cubic foot of syrup per minute. How quickly is the level of the tank decreasing at that time? 5.2 Optimization Problems Problem 13 You want to fit a rectangular solid box inside a circular cone with height 10 feet and base radius 3 feet. It will sit on the base and the upper corners will all touch the diagonal surface of the cone. What are the dimensions of the box with the largest possible volume? Problem 14 You are in the business of designing posters. Typically your customer says something like: “I need a printed area of A with right side margin r, left side margin l, top side margin t and bottom side margin b, and I need it to be as cheap as possible.” Every time, you dust off your calculus skills and figure out the dimensions of the poster with the smallest total area, and your customers are always pleased with the results. Finally, though, you get tired of solving these problems and decide to come up with a formula that will give the perfect dimensions every time, without needing to do the same old problems over and over. Find the required formula! 5.3 Integration Problem 15 You bought a strange vegetable at the market and commenced slicing it thinly as soon as you got home. The slices are pentagons and the length of the side of the pentagonal slice x inches from the left end of the vegetable is given by the expression (x2 +1)(x3 −16x) (it was four inches long before you started cutting it). What was the total volume of the vegetable? Problem 16 An airplane is flying through the air leaving behind a curved 1 5 t + cos(t). If trail of smoke. At time t its speed is v(t) = 1 + t − 13 t3 + 120 the plane starts flying at t = 0 and ends at time t = 8π, then how long is the smoke trail at the end?
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