# 河南快赢481近800开奖结果: Calculus

CHAPTER 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 CHAPTER 2 CHAPTER 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Contents Introduction to Calculus Velocity and Distance Calculus Without Limits The Velocity at an Instant Circular Motion A Review of Trigonometry A Thousand Points of Light Computing in Calculus Derivatives The Derivative of a Function Powers and Polynomials The Slope and the Tangent Line Derivative of the Sine and Cosine The Product and Quotient and Power Rules Limits Continuous Functions Applications of the Derivative Linear Approximation Maximum and Minimum Problems Second Derivatives: Minimum vs. Maximum Graphs Ellipses, Parabolas, and Hyperbolas Iterations x,+ ,= F(x,) Newton s Method and Chaos The Mean Value Theorem and l H8pital s Rule CHAPTER 4 4.1 4.2 4.3 4.4 CHAPTER 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 CHAPTER 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 CHAPTER 7 7.1 7.2 7.3 7.4 7.5 CHAPTER 8 8.1 8.2 8.3 8.4 8.5 Contents The Chain Rule Derivatives by the Chain Rule Implicit Differentiation and Related Rates Inverse Functions and Their Derivatives Inverses of Trigonometric Functions Integrals The Idea of the Integral 177 Antiderivatives 182 Summation vs. Integration 187 Indefinite Integrals and Substitutions 195 The Definite Integral 201 Properties of the Integral and the Average Value 206 The Fundamental Theorem and Its Consequences 213 Numerical Integration 220 Exponentials and Logarithms An Overview 228 The Exponential ex 236 Growth and Decay in Science and Economics 242 Logarithms 252 Separable Equations Including the Logistic Equation 259 Powers Instead of Exponentials 267 Hyperbolic Functions 277 Techniques of Integration Integration by Parts Trigonometric Integrals Trigonometric Substitutions Partial Fractions Improper Integrals Applications of the Integral Areas and Volumes by Slices Length of a Plane Curve Area of a Surface of Revolution Probability and Calculus Masses and Moments 8.6 Force, Work, and Energy Contents CHAPTER 9 9.1 9.2 9.3 9.4 CHAPTER 10 10.1 10.2 10.3 10.4 10.5 CHAPTER 11 11.1 11.2 11.3 11.4 11.5 CHAPTER 12 12.1 12.2 12.3 12.4 CHAPTER 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Polar Coordinates and Complex Numbers Polar Coordinates 348 Polar Equations and Graphs 351 Slope, Length, and Area for Polar Curves 356 Complex Numbers 360 Infinite Series The Geometric Series Convergence Tests: Positive Series Convergence Tests: All Series The Taylor Series for ex, sin x, and cos x Power Series Vectors and Matrices Vectors and Dot Products Planes and Projections Cross Products and Determinants Matrices and Linear Equations Linear Algebra in Three Dimensions Motion along a Curve The Position Vector 446 Plane Motion: Projectiles and Cycloids 453 Tangent Vector and Normal Vector 459 Polar Coordinates and Planetary Motion 464 Partial Derivatives Surfaces and Level Curves 472 Partial Derivatives 475 Tangent Planes and Linear Approximations 480 Directional Derivatives and Gradients 490 The Chain Rule 497 Maxima, Minima, and Saddle Points 504 Constraints and Lagrange Multipliers 514 CHAPTER 14 14.1 14.2 14.3 14.4 CHAPTER 15 15.1 15.2 15.3 15.4 15.5 15.6 CHAPTER 16 16.1 16.2 16.3 Contents Multiple Integrals Double Integrals Changing to Better Coordinates Triple Integrals Cylindrical and Spherical Coordinates Vector Calculus Vector Fields Line Integrals Green s Theorem Surface Integrals The Divergence Theorem Stokes Theorem and the Curl of F Mathematics after Calculus Linear Algebra Differential Equations Discrete Mathematics Study Guide For Chapter 1 Answers to Odd-Numbered Problems Index Table of Integrals CHAPTER 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 CHAPTER 2 CHAPTER 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Contents Introduction to Calculus Velocity and Distance Calculus Without Limits The Velocity at an Instant Circular Motion A Review of Trigonometry A Thousand Points of Light Computing in Calculus Derivatives The Derivative of a Function Powers and Polynomials The Slope and the Tangent Line Derivative of the Sine and Cosine The Product and Quotient and Power Rules Limits Continuous Functions Applications of the Derivative Linear Approximation Maximum and Minimum Problems Second Derivatives: Minimum vs. Maximum Graphs Ellipses, Parabolas, and Hyperbolas Iterations x,+ ,= F(x,) Newton s Method and Chaos The Mean Value Theorem and l H8pital s Rule C H A P T E R 1 Introduction to Calculus 1.4 Velocity and Distance The right way to begin a calculus book is with calculus. This chapter will jump directly into the two problems that the subject was invented to solve. You will see what the questions are, and you will see an important part of the answer. There are plenty of good things left for the other chapters, so why not get started? The book begins with an example that is familiar to everybody who drives a car. It is calculus in action-the driver sees it happening. The example is the relation between the speedometer and the odometer. One measures the speed (or velocity); the other measures the distance traveled. We will write v for the velocity, and f for how far the car has gone. The two instruments sit together on the dashboard: Fig. 1.1 Velocity v and total distance f (at one instant of time). Notice that the units of measurement are different for v and f.The distance f is measured in kilometers or miles (it is easier to say miles). The velocity v is measured in km/hr or miles per hour. A unit of time enters the velocity but not the distance. Every formula to compute v from f will have f divided by time. The central question of calculus is the relation between v and f. --- 1 Introduction to Calculus Can you find v if you know f, and vice versa, and how? If we know the velocity over the whole history of the car, we should be able to compute the total distance traveled. In other words, if the speedometer record is complete but the odometer is missing, its information could be recovered. One way to do it (without calculus) is to put in a new odometer and drive the car all over again at the right speeds. That seems like a hard way; calculus may be easier. But the point is that the information is there. If we know everything about v, there must be a method to find f. What happens in the opposite direction, when f is known? If you have a complete record of distance, could you recover the complete velocity? In principle you could drive the car, repeat the history, and read off the speed. Again there must be a better way. The whole subject of calculus is built on the relation between u and f. The question we are raising here is not some kind of joke, after which the book will get serious and the mathematics will get started. On the contrary, I am serious now-and the mathematics has already started. We need to know how to find the velocity from a record of the distance. (That is called integration goes from v to f. We look first at examples in which these pairs can be computed and understood. CONSTANT VELOCITY Suppose the velocity is fixed at v =60 (miles per hour). Then f increases at this constant rate. After two hours the distance is f =120 (miles). After four hours f =240 and after t hours f =60t. We say that f increases linearly with time-its graph is a straight line. 4 velocity v(t) 4 distancef(t) v 240~~s1~=“=604 Area 240 : I time t time t Fig. 1.2 Constant velocity v =60 and linearly increasing distance f=60t. Notice that this example starts the car at full velocity. No time is spent picking up speed. (The velocity is a “step function.“) Notice also that the distance starts at zero; the car is new. Those decisions make the graphs of v and f as neat as possible. One is the horizontal line v =60. The other is the sloping line f =60t. This v, f, t relation needs algebra but not calculus: if v is constant and f starts at zero then f =vt. The opposite is also true. When f increases linearly, v is constant. The division by time gives the slope. The distance is fl =120 miles when the time is t1 =2 hours. Later f =240 at t, =4. At both points, the ratio f/t is 60 miles/hour. Geometrically, the velocity is the slope o f the distance graph: change in distance -vt slope = -v. change in time t - - 1.1 Velocity and Distance Fig. 1.3 Straight lines f = 20 + 60t (slope 60) and f = -30t (slope -30). The slope of the f-graph gives the v-graph. Figure 1.3 shows two more possibilities: 1. The distance starts at 20 instead of 0. The distance formula changes from 60t to 20 + 60t. The number 20 cancels when we compute change in distance-so the slope is still 60. 2. When v is negative, the graph off goes downward. The car goes backward and the slope off = -30t is v = -30. I don t think speedometers go below zero. But driving backwards, it s not that safe to watch. If you go fast enough, Toyota says they measure “absolute valuesw-the speedometer reads + 30 when the velocity is -30. For the odometer, as far as I know it just stops. It should go backward.? VELOCITY vs. DISTANCE: SLOPE vs. AREA How do you compute f from v? The point of the question is to see f = ut on the graphs. We want to start with the graph of v and discover the graph off. Amazingly, the opposite of slope is area. The distance f is the area under the v-graph. When v is constant, the region under the graph is a rectangle. Its height is v, its width is t, and its area is v times t. This is integration, to go from v to f by computing the area. We are glimpsing two of the central facts of calculus. 1A The slope of the f-graph gives the velocity v. The area under the v-graph gives the distance f. That is certainly not obvious, and I hesitated a long time before I wrote it down in this first section. The best way to understand it is to look first at more examples. The whole point of calculus is to deal with velocities that are not constant, and from now on v has several values. EXAMPLE (Forward and back) There is a motion that you will understand right away. The car goes forward with velocity V, and comes back at the same speed. To say it more correctly, the velocity in the second part is -V. If the forward part lasts until t = 3, and the backward part continues to t = 6, the car will come back where it started. The total distance after both parts will be f = 0. +This actually happened in Ferris Bueller s Day 08,when the hero borrowed his father s sports car and ran up the mileage. At home he raised the car and drove in reverse. I forget if it worked. 1 Introductionto Calculus 1u(r) = slope of f(t) Fig. 1.4 Velocities + V and -V give motion forward and back, ending at f(6)=0 . The v-graph shows velocities + V and -V. The distance starts up with slope + V and reaches f = 3V. Then the car starts backward. The distance goes down with slope -V and returns to f = 0 at t = 6. Notice what that means. The total area “under“ the v-graph is zero! A negative velocity makes the distance graph go downward (negative slope). The car is moving backward. Area below the axis in the v-graph is counted as negative. FUNCTIONS This forward-back example gives practice with a crucially important idea-the con- cept of a “jiunction.“ We seize this golden opportunity to explain functions: The number v(t) is the value of the function t. at the time t. The time t is the input to the function. The velocity v(t) at that time is the output. Most people say “v oft“ when they read v(t). The number “v o f 2“ is the velocity when t = 2. The forward-back example has v(2) = + V and v(4) = -V. The function contains the whole history, like a memory bank that has a record of v at each t. It is simple to convert forward-back motion into a formula. Here is v(t): The ,right side contains the instructions for finding v(t). The input t is converted into the output + V or -V. The velocity v(t) depends on t. In this case the function is “di~continuo~s,~ because the needle jumps at t = 3. The velocity is not dejined at that instant. There is no v(3). (You might argue that v is zero at the jump, but that leads to trouble.) The graph off has a corner, and we can t give its slope. The problem also involves a second function, namely the distance. The principle behind f(t) is the same: f (t) is the distance at time t. It is the net distance forward, and again the instructions change at t = 3. In the forward motion, f(t) equals Vt as before. In the backward half, a calculation is built into the formula for f(t): At the switching time the right side gives two instructions (one on each line). This would be bad except that they agree: f (3)= 3V.vhe distance function is “con- ?A function is only allowed one ~:alue,f (r) at each time ror ~ ( t ) 1.1 Velocity and Distance tinuous.“ There is no jump in f, even when there is a jump in v. After t = 3 the distance decreases because of -Vt. At t = 6 the second instruction correctly gives f (6) = 0. Notice something more. The functions were given by graphs before they were given by formulas. The graphs tell you f and v at every time t-sometimes more clearly than the formulas. The values f (t) and v(t) can also be given by tables or equations or a set of instructions. (In some way all functions are instructions-the function tells how to find f at time t.) Part of knowing f is knowing all its inputs and outputs-its domain and range: The domain of a function is the set of inputs. The range is the set of outputs. The domain of f consists of all times 0 0). Then draw (b) U(t) + 2 (4 U(t + 2) ( 4 3UW (e) U(3t). 45 (a) Draw the graph of f (t) = t + 1 for -1 Q t 6 1. Find the domain, range, slope, and formula for (b) 2f (0 (4 f (t -3) (d) -f (0 (el f kt). 46 If f (t) = t -1 what are 2f (3t) and f (1 -t) and f (t -I)? 47 In the forward-back example find f (* T)and f(3T). Verify that those agree with the areas “under“ the v-graph in Figure 1.4. 48 Find formulas for the outputs fl(t) and fi(t) which come from the input t: (1) inside = input * 3 (2) inside +input + 6 output = inside + 3 output tinside* 3 Note BASIC and FORTRAN (and calculus itself) use = instead of t.But the symbol tor := is in some ways better. The instruction t + t + 6 produces a new t equal to the old t plus six. The equation t = t + 6 is not intended. 49 Your computer can add and multiply. Starting with the number 1 and the input called t, give a list of instructions to lead to these outputs: f1(t)=t2+t f2(t)=fdfdt)) f3(t)=f1(t+l)- 50 In fifty words or less explain what a function is. The last questions are challenging but possible. 51 If f (t) = 3t -1 for 0 6 t Q 2 give formulas (with domain) and find the slopes of these six functions: (a) f (t + 2) (b) f(t) + 2 (4 2f (0 ( 4 f (2t) (e) f (-t) (f) f (f (t)). 52 For f (t) = ut + C find the formulas and slopes of (a) 3f (0 + 1 (b) f(3t + 1) (c) 2f(4t) (dl f (-t) (el f (0 -f (0) (f) f (f (t)). 53 (hardest) The forward-back function is f (t) = 2t for O0. 1.2 Calculus Without Limits 13 What is the slope of the step function? It is zero except at the jump. At that moment, which is t = 0, the slope is infinite. We don t have an ordinary velocity v(t)-instead we have an impulse that makes the car jump. The graph is a spike over the single point t = 0, and it is often denoted by 6-so the slope of the step function is called a “delta function.“ The area under the infinite spike is 1. You are absolutely not responsible for the theory of delta functions! Calculus is about curves, not jumps. Our last example is a real-world application of slopes ands rates-to explain “how taxes work.“ Note especially the difference between tax rates and tax brackets and total tax. The rates are v, the brackets are on x, the total tax is f. EXAMPLE 3 Income tax is piecewise linear. The slopes are the tax rates .15,.28,.31. Suppose you are single with taxable income of x dollars (Form 1040, line 37-after all deductions). These are the 1991 instructions from the Internal Revenue Service: If x is not over $20,350, the tax is 15% of x. If $20,350 $49,300 pays tax at the top rate of 31%. But only the income in that bracket is taxed at that rate. Figure 1.11 shows the rates and the brackets and the tax due. Those are not average rates, they are marginal rates. Total tax divided by total income would be the average rate. The marginal rate of.28 or .31 gives the tax on each additional dollar of income- it is the slope at the point x. Tax is like area or distance-it adds up. Tax rate is like slope or velocity-it depends where you are. This is often unclear in the news media. A? 1 on - .U IO sup 180 =slope60 11,158- across 3 f(2)= 40 S? slpe 20 3,052- k tax to pay f(x) 31% tax rate = slope .28 15% taxable income I I Y 2 5 2 5 20,350 49,300 Fig. 1.11 The tax rate is v, the total tax is f. Tax brackets end at breakpoints. Question What is the equation for the straight line in the top bracket? Answer The bracket begins at x = $49,300 when the tax is f(x) = $11,158.50. The slope of the line is the tax rate .31. When we know a point on the line and the slope, we know the equation. This is important enough to be highlighted. Section 2.3 presents this “point-slope equation“ for any straight line. Here you see it for one specific example. Where does the number $11,158.50 come from? It is the tax at the end of the middle bracket, so it is the tax at the start of the top bracket. v2 = 60 ov = 20 1 Introduction t o Calculus Figure 1.11 also shows a distance-velocity example. The distance at t = 2 is f (2)= 40 miles. After that time the velocity is 60 miles per hour. So the line with slope 60 on the f-graph has the equation f (t) = starting distance + extra distance =40 + 60(t -2). The starting point is (2 40). The new speed 60 multiplies the extra time t -2. The point-slope equation makes sense. We now review this section, with comments. Central idea Start with any numbers in f. Their differences go in v. Then the sum of those differences is ha,,-ffirst. Subscript notation The numbers are f,, fl, . and the first difference is v, =fl-f,. A typical number is fi and the jth difference is vj =fi -fi-. When those differences are added, all f s in the middle (like f,) cancel out: Examples fi =j or j2or 2 . Then vj = 1 (constant) or 2j -1 (odd numbers) or 2 - . Functions Connect the f s to be piecewise linear. Then the slope v is piecewise constant. The area under the v-graph from any t,,,,, to any ten, equals f (ten,)-f (t,,,,,). Units Distance in miles and velocity in miles per hour. Tax in dollars and tax rate in (dollars paid)/(dollars earned). Tax rate is a percentage like .28, with no units. 1.2 EXERCISES Read-through questions Start with the numbers f = 1,6,2,5. Their differences are v = a .The sum of those differences is b .This is equal to f , , , , minus c . The numbers 6 and 2 have no effect on this answer, because in (6 -1)+ (2 -6) + (5 -2) the numbers 6 and 2 d . The slope of the line between f(0) = 1 and f (1) = 6 is e . The equation of that line is f (t) = f . With distances 1, 5, 25 at unit times, the velocities are g . These are the h of the f-graph. The slope of the tax graph is the tax i . If f(t) is the postage cost for t ounces or t grams, the slope is the i per k . For distances 0, 1,4,9 the velocities are I . The sum of the first j odd numbers is fi = m . Then flo is n and the velocity ulo is 0 . The piecewise linear sine has slopes P . Those form a piecewise q cosine. Both functions have r equal to 6, which means that f (t + 6) = s for every t. The veloci- ties v = 1,2,4,8, . have vj = t . In that case fo = 1 and j j . = u . The sum of 1,2,4,8, 16 is v . The difference 2J -2 - equals w . After a burst of speed V to time T, the distance is x . If f(T) = 1 and V increases, the burst lasts only to T = Y . When V approaches infinity, f (t) approaches a function. The velocities approach a A function, which is concentrated at t = 0 but has area B under its graph. The slope of a step function is c . Problems 1-4 are about numbers f and differences v. 1 From the numbers f = 0,2,7,10 find the differences u and the sum of the three v s. Write down another f that leads to the same v s. For f= 0,3,12,10 the sum of the u s is still . 2 Starting