Math 221 – 1st Semester Calculus Lecture Notes for Fall 2006. Prof J. Robbin December 21, 2006 All references to “Thomas” or “the textbook” in these notes refer to Thomas’ CALCULUS 11th edition published by Pearson Addison Wesley in 2005. These notes may be downloaded from http://www.math.wisc.edu/~robbin/221dir/lecs-221.pdf. Some portions of these notes are adapted from http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/. Some problems come from a list compiled by Arun Ram, others come from the WES program, and others come from the aforementioned Thomas text or from Stewart, Calculus (Early Transcendentals), 3rd Edition. 1 Contents I What you need to know to take Calculus 221 5 1 Algebra 5 2 Coordinate Geometry 7 3 Functions 9 4 Trigonometry 12 5 Additional Exercises 15 II Limits 17 6 Tangent and Velocity 17 7 Limits 19 8 Two Limits in Trigonometry 26 9 Continuity 27 III Differentiation 31 10 Derivatives Defined 31 11 Higher Derivatives and Differential Notation 37 12 Implicit Functions 41 13 The Chain Rule 43 14 Inverse Functions 47 15 Differentiating Trig Functions 50 16 Exponentials and Logarithms 53 17 Parametric Equations 59 18 Approximation* 63 19 Additional Exercises 67 2 IV Applications of Derivatives 68 20 The Derivative as A Rate of Change 68 21 Related Rates 71 22 Some Theorems about Derivatives 74 23 Curve Plotting 81 24 Max Min Word Problems 84 25 Exponential Growth 87 26 Indeterminate Forms (l’Hˆopital’s Rule) 91 27 Antiderivatives 93 28 Additional Problems 95 V Integration 96 29 The Definite Integral 96 30 The Fundamental Theorem of Calculus 104 31 Averages 107 32 Change of Variables 108 VI Applications of Definite Integrals 112 33 Plane Area 112 34 Volumes 114 35 Arc Length 118 36 Surface Area 122 37 Center of Mass 125 VII Loose Ends 132 38 The Natural Log Again* 132 3 39 Taylor Approximation* 133 40 Newton’s Method* 135 VIII More Problems 136 IX Notes for TA’s 153 4 Chapter I What you need to know to take Calculus 221 In this chapter we will review material from high school mathematics. We also teach some of this material at UW in Math 112, Math 113, and Math 114, and some of it will be reviewed again as we need it. This material should be familiar to you. If it is not, you may not be ready for calculus. Pay special attention to the definitions. Important terms are shown in boldface when they are first defined. 1 Algebra This section contains some things which should be easy for you. (If they are not, you may not be ready for calculus.) §1.1. Answer these questions. 1. Factor x2 − 6x + 8. 2. Find the values of x which satisfy x2 − 7x + 9 = 0. (Quadratic formula.) 3. x2 − y2 =? Does x2 + y2 factor? √ 4. True or False: x2 + 4 = x + 2? √ 5. True or False: (9x)1/2 = 3 x? x2x8 6. True or False: = x2+8−3 = x7? x3 7. Find x if 3 = log2(x). x 8. What is log7(7 )? 9. True or False: log(x + y) = log(x) + log(y)? 10. True or False: sin(x + y) = sin(x) + sin(y)? §1.2. There are conventions about the order of operations. For example, ab + c means (ab) + c and not a(b + c), a b means a/(b/c) and not (a/b)/c, c a b means (a/b)/c and not a/(b/c), c log a + b means (log a) + b and not log(a + b). If necessary, we use parentheses to indicate the order of doing the operations. 5 §1.3. There is analogy between the laws of addition and the laws of multipli- cation: a + b = b + a ab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) a + 0 = a a · 1 = a a + (−a) = 0 a · a−1 = 1 a − b = a + (−b) a/b = a · b−1 a − b = (a + c) − (b + c) a/b = (ac)/(bc) (a − b) + (c − d) = (a + c) − (b + d) (a/b) · (c/d) = (ac)/(bd) (a − b) − (c − d) = (a + d) − (b + c) (a/b)/(c/d) = (ad)/(bc) The last line explains why we invert and multiply to divide fractions. The only other law of arithmetic is the distributive law (a + b)c = ac + bc, c(a + b) = ca + cb. Note that (a + b)/c = (a/c) + (b/c), but1 c/(a + b) 6= (c/a) + (c/b). Exercises Exercise 1.4. Answer the questions in §1.1 Exercise 1.5. Here is a list of some algebraic expressions that have been “sim- plified.” Some steps in the simplification processes are correct and some of them are WRONG! For each problem: 1. Determine if the simplified result is correct. 2. Determine if there are any mistakes made in the simplication process. (NOTE: just because the result is correct does not mean there are no mistakes). 3. If there are mistakes, redo the problem correctly. If there are no mistakes, redo the problem with another correct method. x2 − 1 x2 + (−1) x2 −1 (i) = = + = x − 1 x + 1 x + 1 x 1 (ii)(x + y)2 − (x − y)2 = x2 + y2 − x2 − y2 = 0 1usually 6 9(x − 4)2 32(x − 4)2 (3x − 12)2 (iii) = = = 3x − 12 3x − 12 3x − 12 3x − 12 x2y5 (iv) = x2y5 · 2x3 = 2x6y5 2x−3 (2x3 + 7x2 + 6) − (2x3 − 3x2 − 17x + 3) 4x2 − 17x + 9 9 (v) = = 2x − 17 + (x + 8) + (x − 8) 2x 2x x−1 + y−1 (x + y)−1 x + y −1 x + y x + y (vi) = = = − = . x−1 − y−1 (x − y)−1 x − y x − y y − x 2 Coordinate Geometry The material in this section is crucial for understanding calculus. It is reviewed in Thomas pages 9-18, but if you have not seen it before you may not be ready for calculus. §2.1. Points and Slope. The notation P (x, y) is used as an abbreviation for the more cumbersome phrase “the point P whose coordinates are (x, y).” The slope of the line through the distinct points P0(x0, y0) and P1(x1, y1) is y − y m = 1 0 . y x1 − x0 6 P1 ¨¨ ¨¨ ¨ y1 − y0 ¨ ¨¨ P0¨ x − x 1 0 - x §2.2. Point-Slope Equation of a Line. If we use a different pair of points on the line to compute the slope, we get the same answer. Hence, a point P (x, y) lies on the line P0P1 if and only if we get the same answer for the slope when we use (x, y) in place of (x1, y1): y − y y − y 0 = 1 0 = m. x − x0 x1 − x0 This equation has one minor flaw; it doesn’t work when (x, y) = (x0, y0) (never divide by zero). To remedy this multiply by (x − x0) and add y0 to both sides: 7 y = y0 + m(x − x0). This is the equation for the line through P0(x0, y0) with slope m; this form makes it obvious that the point P0(x0, y0) lies on the line. For example, the equation for the line through P0(2, 3) and P1(4, 11) is y − 3 11 − 3 = = 4, or y = 3 + 4(x − 2), x − 2 4 − 2 See Thomas page 11. §2.3. If a line has slope m, then any line perpendicular to it has slope −1/m. (Proof: The slope of a line is the tangent of the angle it makes with the x-axis π and tan(φ + 2 ) = −1/ tan(φ). §2.4. Sometimes we use letters other than x and y. For this reason, always label the axes when you draw a graph. For example the line 2u + 3v = 1 can 2 1 3 1 be written as v = − 3 u + 3 and as u = − 2 v + 2 . If the u-axis is horizontal, the 2 3 slope is − 3 . If the v-axis is horizontal, the slope is − 2 . The slope is always the “rise over the run”, i.e. to find the slope, take two points on the line, subtract the vertical coordinates (this is the “rise”), subtract the horizontal coordinates (this is the “run”), and divide the rise by the run. §2.5. Distance and Circles. By the Pythagorean theorem the distance from P0(x0, y0) to P1(x1, y1) is p 2 2 |P0P1| = (x1 − x0) + (y1 − y0) The distance from a point P (x, y) to the point P0(x0, y0) is exactly a when 2 2 |P0P | = a , i.e. 2 2 2 (x − x0) + (y − y0) = a . This is the equation for the circle centered at P0 with radius a. For example, the equation of the circle of radius 5 centered at P0(2, 3) is (x − 2)2 + (y − 3)2 = 52, or x2 + y2 − 4x − 6y − 12 = 0. The latter equation has the form of a general 2nd degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 (with A = C = 1, B = 0, D = −4, E = −6, F = −12). Ellipses (including circles), parabolas, and hyperbolas all have equations of this form. 8 Exercises Exercise 2.6. Let L1 and L2 be two lines. An equation for L1 is 6x + 4y = 7. The line L2 is perpendicular to L1 and goes through the point P0(−1, 0). Find an equation for the line L2 and the point where the two lines intersect. Exercise 2.7. Repeat the previous exercise replacing 6x+4y = 7 by ax+by = c and P0(−1, 0) by P0(x0, y0).