Homework

Homework is to be turned in at the beginning of the discussion section on the due date. It will be graded and returned to you in your discussion section.
FFT = "Food for thought" (problems that you should think about, but not turn them in with the regular homework)
Click on an underlined problem to see a hint.
Abbreviations: RevCh4 = Exercises in Review for Ch. 4 (pages 338-340); CC4 = Concept Check for Ch. 4 (page 337); T/FQ4 = True/False Quiz for Ch. 4 (page 338); RevCh5 = Exercises in Review for Ch. 5 (pages 378-379); CC5 = Concept Check for Ch. 5 (page 377); CC6 = Concept Check for Ch. 6 (pages 480-481); T/FQ6 = True/False Quiz for Ch. 6 (page 481); RevCh6 = Exercises in Review for Ch. 6 (pages 481-484); CC7 = Concept Check for Ch. 7 (page 553 ); T/FQ7 = True/False Quiz for Ch. 7 (page 554); RevCh7 = Exercises in Review for Ch. 7 (pages 554-556)

Content of the lectures

• Lecture 1 (Mon, Jan 12): Sigma notation: definition, examples; basic properties (Theorem 2); sum of n 1's (Example 3); formula for the sum of the natural numbers from 1 to n (Example 4); formula for the sum of the squares of the natural numbers from 1 to n (Example 5) - read only Solution 1 (using telescoping sum); Theorem 3; Example 6 [Appendix E, skip Solution 2 of Example 5, and Example 7]
• Lecture 2 (Wed, Jan 14): Areas and distances: approximating the area under a graph of a function ƒ from x=a to x=b by sum of the areas of rectangles; using left and right endpoints to to find approximations (L_n and R_n) of the area under the graph of ƒ; Example: computing the area under a graph of the function ƒ(x)=x from x=0 to x=1 as a limit as n→∞ of the corresponding R_n; Example 2: computing the area under a graph of the function ƒ(x)=x² from x=0 to x=1 as a limit as n→∞ of the corresponding R_n [pages 284-288 of Sec. 4.1]
• Lecture 3 (Fri, Jan 16): Areas and distances (cont.): more general choices of sample points x_i^* (at which the value ƒ(x_i^*) of the function ƒ is calculated and used as the height of a rectangle); computing the traveled distance as the area under the graph of the speed as a function of time by using approximation by rectangles [pages 289-293 of Sec. 4.1]
  The definite integral: definition of a definite integral as a limit of sums; sample points, integrable functions, integral sign, integrand, upper and lower limits of integration, integration (the process of computing the value of an integral); Riemann sums; net area; functions that are continuous or have a finite number of jump discontinuities are integrable (theorem 3); writing a definite integral over the interval [a,b] as a limit of Riemann sums (Theorem 4); Examples 2-4 of computing integrals [pages 295-301 of Sec. 4.2]
• Lecture 4 (Wed, Jan 21): The definite integral (cont.): the Midpoint Rule; properties of the definite integral: switching sign when switching limits of integration, integral of a constant function, linearity property, combining integrals over adjacent intervals, comparison properties; examples [pages 302-306 of Sec. 4.2]
• Lecture 5 (Fri, Jan 23): The Fundamental Theorem of Calculus: motivation and sketch of the proof of the Part 1 of the Fundamental Theorem of Calculus (FTC), examples; Part 2 of the FTC, examples [pages 310-313, 315-317 of Sec. 4.3; skip Example 4]
• Lecture 6 (Mon, Jan 26): The Fundamental Theorem of Calculus (cont.): examples of using the FTC and the Chain Rule to find the derivative of a definite integral whose limits of integrations are functions of x [Example 4 on page 314 of Sec. 4.3]
  Indefinite integrals and the net change theorem: indefinite integral of a function (same as antiderivative); making a table of indefinite integrals by using a table of derivatives; using indefinite integrals and the FTC to compute definite integrals [pages 321-323 of Section 4.4]
• Lecture 7 (Wed, Jan 28): Indefinite integrals and the net change theorem (cont.): rate of change F'(t)=dF(t)/dt of a quantity F(t) depending on the time t; FTC and the Net Change Theorem; examples of the rate of change of:
  • the volume V(t) of water in a container,
  • the concentration [C](t) of a substance,
  • the mass m(x) of a rod between the points with coordinates 0 and x (with m'(t)=ρ(x), the linear density of the rod),
  • the population n(t),
  • the position s(t) (with s'(t)=v(t), the velocity),
  • the total traveled distance (with its derivative equal to |v(t)|, the speed of motion),
  • the velocity (with v'(t)=a(t), the acceleration)
  [pages 323-326 of Section 4.4]
• Lecture 8 (Fri, Jan 30): The Substitution Rule: the Substitution Rule for indefinite integrals; justification of the Substitution Rule (it follows from the Chain Rule); the Substitution Rule for definite integrals; examples [pages 330-334 of Sec. 4.5]

• Lecture 9 (Mon, Feb 2): The Substitution Rule (cont.): more examples of applying the Substitution Rule; definite integrals of even and of odd functions over symmetric intervals (i.e., intervals of the form [−a,a]); examples [pages 334-335 of Sec. 4.5]
  Areas between curves: expressing the area between two graphs, y=ƒ(x) and y=g(x), as a definite integral of |ƒ(x)−g(x)|; examples [pages 344-347 of Sec. 5.1]
  

• Lecture 10 (Wed, Feb 4): Areas between curves (cont.): considering x as a function of y and expressing the area between two graphs, x=ƒ(y) and x=g(y), as a definite integral of |ƒ(y)−g(y)|; examples [page 348 of Sec. 5.1]
  Volumes: "slicing" the solid by many planes perpendicular to the x-axis, and replacing each "slice" with a cylinder with vertical walls with area of the base A(x_i^*) and height Δx, i.e., with volume ΔV_i = A(x_i^*)Δx; the total volume of the solid is the limit as n→∞ of the Riemann sum of terms  A(x_i^*)Δx over i from 1 to n; using the definition of definite integral, the total volume of the solid extending from x=a to x=b is integral from a to b of the function A(x); Example 1 (computing the volume of a ball of radius r) [pages 352-354 of Sec. 5.2]
  

• Lecture 11 (Fri, Feb 6): Volumes (cont.): computing the volume of a (straight circular) cylinder of height H and radius of the base R; computing the volume of ball of radius R; more examples (read Examples 2-4, 7) [pages 355, 356, 357, 358 of Sec. 5.2]
  Volumes by cylindrical shell: representing a solid of revolution (i.e., obtained by rotating a plane region about a line) as a limit of the Riemann sum of the volumes of thin cylindrical shells; computing the volume of a thin cylindrical shell of inner radius r, wall thickness Δx, and height h in two ways: exactly, ΔV=[2πrΔx+π(Δx)²]h, and approximately, ΔV≈2πr(Δx)h; in the derivation of the approximate expression for ΔV we used that the area between the concentric circles with radii r and (r+Δx) is approximately equal to the area of a rectangle with sides 2πr (the circumference of the inner circle) and Δx ("straighening up" the side wall of a cylindrical can); derivation of the expression for the volume of a solid of revolution obtained by rotating the area under y=ƒ(x) from x=a to x=b around the x-axis as the integral over x from a to b of 2πxƒ(x) [pages 363-364 of Sec. 5.3]
• Lecture 12 (Mon, Feb 9): Volumes by cylindrical shell (cont.): examples [Examples 1-4 on pages 365-366 of Sec. 5.3]
  Challenge: set up the integrals for the volume of the solid torus by slicing (Exercise 5.2/61) and by cylindrical shells (Exercise 5.3/46). Click here for hints. This is not a homework problem, it do not turn it in!
  Work: "high school" definition of work of a constant force: (Work W)=(Force F)×(distance x); the work of a force F(x) that depends on the coordinate x when the object is displaced by Δx is approximately ΔW≈F(x)Δx; the work of the force F(x) when the object is displaced from x=a to x=b is equal to the limit of the Riemann sum of terms F(x_i^*)Δx, i.e., to the integral of F(x) over x from a to b; Hooke's law: the elastic force due to a stretched spring is F(x)=kx (with x=0 chosen at the equilibrium position); computation of the work done by an elastic force; energy conversion between kinetic (the energy of the motion) and potential (the "stored" energy of the interaction) in oscillations of a mass attached to a spring; read Example 4 (computing the work done in lifting a hanging rope to the top of a building) [pages 368-370 of Section 5.4]
• Lecture 13 (Wed, Feb 11): Work (cont.): computing the work needed to pump all water out of a conical tank (similarly to Example 5, but with the cone upside down): constructing the Riemann sum of the small amounts of work needed to lift the thin "slices" of water to the level of the spout, representing this as a definite integral, and solving the integral; checking the units of the physical quantities as a method for detecting errors [pages 370-371 of Sec. 5.4]
  Average value of a function: discussion of averaging in calculating the average grade of all students in two Calculus sections of different size - "weighted average"; meaning of average value of a function [pages 373-374 of Sec. 5.5]
• Lecture 14 (Fri, Feb 13): Exam 1 [on Sections 4.1-4.5, 5.1, 5.2 and App E, covered in Lectures 1-11]
• Lecture 15 (Mon, Feb 16): Average value of a function (cont.): discussion of the importance of average in reporting statistical data; median vs. average; Mean Value Theorem for Integrals - statement and discussion of the importance of the requirement of continuity of ƒ [pages 374-375 of Sec. 5.5]
• Lecture 16 (Wed, Feb 18): Average value of a function (cont.): Mean Value Theorem for Integrals - proof of the theorem based on the Mean Value Theorem for Derivatives; exercise (not to be turned in): prove that if m≤ƒ(x)≤M for all x∈[a,b], then the average ƒ_ave of ƒ over the interval [a,b] satisfies m≤ƒ_ave≤M [exercise 23 of Sec. 5.5 on page 376]
  Inverse functions: domain and range of a function; one-to-one functions; horizontal line test (whether a function is one-to-one); inverse function of a one-to-one function; cancellation equations; examples; derivatives of inverse functions [page 10 of Sec. 1.1; Sec. 6.1]
• Lecture 17 (Fri, Feb 20): Inverse functions (cont.): Mock Quiz 5.
  Exponential functions and their derivatives: let a=const>0; defining the exponential function aⁿ=aa⋅⋅⋅a (k times) for n positive integer; fundamental property: aⁿa^k=a^n+k for any positive integers n and k; "promoting" the fundamental property to a requred property of the general exponential function: a^xa^y=a^x+y for any real numbers x and y; defining a⁰=1 and a⁻ⁿ=1/aⁿ for n positive integer; defining the exponential functions for rational numbers a^n/k as the kth root of aⁿ; defining the exponential functions for irrational numbers as a limit by requiring that the exponential function be continuous; properties of the exponential function: a^xa^y=a^x+y, a^x/a^y=a^x−y, (a^x)^y=a^xy, (ab)^x=a^xb^x; limits of a^x as x→∞ and x→−∞ for a∈(0,1) and for a>1; proof that if ƒ(x)=a^x, then ƒ'(x)=ƒ'(0)ƒ(x); defining the number e as the (only) number such that if ƒ(x)=e^x, then ƒ'(0)=1, i.e., (e^x)'=e^x; indefinite integral of e^x [pages 391-397, 400 of Sec. 6.2]
• Lecture 18 (Mon, Feb 23): Exponential functions and their derivatives (cont.): Mock Quiz 6.
  Logarithmic functions: definition of the logarithmic function log_a as the inverse function to the exponential function a^x for a∈(0,1) or a>(0,∞) through the cancellation equations log_a(a^x)=x for any x∈R or, equivalently, a^log_ay=y for any y∈(0,∞); properties of log_a: log_a(xy)=log_ax+log_ay, log_a(x/y)=log_ax−log_ay, log_a(x^r)=r log_ax; limits of log_ax for x→±∞ for a∈(0,1) and for a>(0,∞); natural logarithm, ln=log_e; change of base formula: log_ax=(ln x)/(ln a); examples [pages 404-407 of Sec. 6.3]
• Lecture 19 (Wed, Feb 25): Derivatives of logarithmic functions: derivative of ln(x) and of ln|x|; integral of 1/x; derivatives and integrals of general logarithmic and exponential functions; logarithmic differentiation; examples [pages 410-417 of Sec. 6.4]
• Lecture 20 (Fri, Feb 27): Inverse trigonometric functions: making the function sin one-to-one by restricting its domain to [−π/2,π/2]; defining the inverse function of sin, arcsin=sin⁻¹:[−1,1]→[−π/2,π/2]; derivation of the formula for the derivative of arcsin: dsin⁻¹x/dx=(1−x²)^−1/2; making the function cos one-to-one by restricting its domain to [0,π]; defining the inverse function of cos, arccos=cos⁻¹:[−1,1]→[0,π]; derivative of arccos: dcos⁻¹x/dx=−(1−x²)^−1/2; Examples 5(a), 7 [pages 453-455, 458-459 of Sec. 6.6]
  Reading assignment (optional): the number e as a limit; compute the numerical value of (1+(1/n))ⁿ for n=10⁶,10⁷,10⁸,10⁹,10¹⁰ on your calculator, think about the roundoff error of the calculation [pages 417-418 of of Sec. 6.4]
  

• Lecture 21 (Mon, Mar 2): Inverse trigonometric functions (cont.): making the function tan one-to-one by restricting its domain to (−π/2,π/2); defining the inverse function of tan, arctan=tan⁻¹:R→(−π/2,π/2); derivation of the formula for the derivative of arctan: dtan⁻¹x/dx=1/(1+x²); Examples 5(b), 6, 8, 9 [pages 456-459 of Sec. 6.6]
  Hyperbolic functions: definition of cosh, sinh, and tanh; basic facts about hyperbolic functions: cosh is even, sinh and tanh are odd, cosh²x−sinh²x=1, 1−tanh²x=cosh⁻²x; derivatives of cosh, sinh, and tanh; definition of the inverse hyperbolic functions arsinh=sinh⁻¹, arcosh=cosh⁻¹, artanh=tanh⁻¹; exercise: use the definition, y=sinh(y)=(e^x−e^−x)/2, to express z=e^x in terms of y and prove that arsinh(y)=ln(y+(y²+1)^1/2) [pages 462-465 of Sec. 6.7]
  
• Lecture 22 (Wed, Mar 4): Lecture canceled due to weather.
• Lecture 23 (Fri, Mar 6): Hyperbolic functions (cont.): derivatives of the inverse hyperbolic functions arsinh, arcosh, and artanh; two derivations of the derivative of sinh⁻¹; examples [pages 465-467 of Sec. 6.7]
  Indeterminate forms and l'Hospital's rule: l'Hospital's rule for indeterminate forms of the form 0/0 and ∞/∞; examples; applying l'Hospital's rule to the following cases:
  • indeterminate products of the form 0⋅∞ by writing them in the form 0⋅∞=0/(1/∞)=0/0 or ∞⋅0=∞/(1/0)=∞/∞;
  • indeterminate differences of the form ∞−∞ by converting them to the form 0/0 or ∞/∞;
  • indeterminate powers ƒ(x)^g(x) of the form 0⁰, ∞⁰, 1^∞ by writing them in as e^{g(x) ln ƒ(x)} and applying l'Hospital's rule to the lindeterminate product g(x) ln ƒ(x);
  [pages 469-475 of Sec. 6.8]
  Remark: You do not need to memorize the expressions for the derivatives of sinh⁻¹, cosh⁻¹, and tanh⁻¹ - in the exam these expressions will be given to you, and you will only need to know how to use them in problems.
• Lecture 24 (Mon, Mar 9): Indeterminate forms and l'Hospital's rule (cont.): more examples [pages 473-475 of Sec. 6.8]
  Integration by parts: derivation of the formula for integration by parts from the product rule [page 488 of Sec. 7.1]
• Lecture 25 (Wed, Mar 11) Just for fun: famous curves - the shape of a freely hanging chain is a curve called a catenary and is described by the hyperbolic cosine function (see Stewart, page 463 and Exercises 6.7/50 and 6.7/52 on page 468); the 'brachistochrone', i.e., the curve between two points that is such that the time it takes a bead sliding without friction on this curve to get from one end of the curve to the other, is a cycloid (a curve traced by a point on the periphery of a wheel that is rolling without friction on a horizontal plane - see Stewart, pages 663, 664 of Sec. 10.1).
  Integration by parts (cont.): examples: integrating by parts x sin x,  ln x,  x²e^x,  e^x sin x,  tan⁻¹x [pages 488-491 of Sec. 7.1]
• Lecture 26 (Fri, Mar 13): Exam 2 [on Sections 5.3-5.5, 6.1-6.4, 6.6-6.8, covered in Lectures 12, 13, 15-21, 23]
• Lecture 27 (Mon, Mar 23): Integration by parts (cont.): an example: integration by parts of xe^3x and a detailed discussion why setting u=x, dv=d(e^3x/3)=e^3xdx is good, while the substitution u=e^3x, dv=d(x²/2)=xdx is bad; deriving a recursive formula for the integral I_n of xⁿe^x: I_n=xⁿe^x−n I_n−1; using the recursive formula for I_n in order to find I₃ [pages 491-492 of Sec. 7.1 (read Example 6); Exercises 7.1/52 and 7.1/56]
  Trigonometric integrals: integrating sin^mx cosⁿx when at least one of the integers m and n is odd; food for thought: the expression sin⁹⁵x cos⁵x can be integrated in two ways - by setting u=cos x and rewriting sin⁹⁵x cos⁵x dx as u⁹⁵(1−u²)² du, or by setting v=sin x and rewriting sin⁹⁵x cos⁵x dx as −(1−v²)⁴⁷v⁵ dv - which substitution is more convenient (a lot more convenient)? [Examples 1 and 2 on pages 495, 496; parts (a) and (b) of the Strategy box on page 497 of Sec. 7.2]
• Lecture 28 (Wed, Mar 25): A digression: All of trigonometry!: deriving all trigonometric relations from the formulas for sine/cosine of a sum of two angles: sin(α+β)=sin α cos β + cos α sin β,   cos(α+β)=cos α cos β − sin α sin β ; deriving formulas for sin(α−β), cos(α−β), sin(2α), cos(2α), tan(α+β) (challenge: derive an expression for sin(3α) and sin(4α) in terms of only sin α and cos α); representing sin α cos β as a sum of two sines/cosines (by adding/subtracting the formulas for sin(α+β) and sin(α−β)), challenge: derive similar expressions for sin α sin β and cos α cos β (these expressions are given in the box on page 500 of the book).
  Trigonometric integrals (cont.): integrating sin^mx cosⁿx when both the integers m and n are even - see part (c) of the Strategy box on page 497 and Examples 3 and 4; tricks for integrating tan^mx secⁿx: use the facts that (tan x)'=sec²x, (sec x)'=tan x sec x, and the identity 1+tan²x=sec²x - see the Strategy box on page 498 and Examples 5 and 6; integrating tan x (challgenge: how about integral of cot x?), csc x, and sec x, Examples 7 and 8; integrating a product of sines and cosines by representing it as a sum of sines and cosines - see the Strategy box on page 500 and Example 9 [pages 497-500 of Sec. 7.2]
• Lecture 29 (Fri, Mar 27): Trigonometric substitution: basic identities used in trigonometric substitution: sin²x+cos²x=1, 1+tan²x=sec²x; three basic trigonometric subsitutions (the table on page 502); Examples 1, 2, 4, 7; substitutions using hyperbolic functions based on the basic identity cosh²x−sech²x=1 - see solution 2 of Example 5 [Sec. 7.3]
• Lecture 30 (Mon, Mar 30): Trigonometric substitution (cont.): more examples of trigonometic and hyperbolic substitution; a detailed solution of Exercise 7.3/32 by using two different substitutions [Sec. 7.3]
  Integration of rational functions by partial fractions: read the handout on partial fractions and Examples 2, 3, 5 [Sec. 7.4]
• Lecture 31 (Wed, Apr 1): Integration of rational functions by partial fractions (cont.): more examples of using partial fractions (Examples 6, 7, 8); useful tricks from elementary algebra: a²−b²=(a−b)(a+b), a³−b³=(a−b)(a²+ab+b²), a³+b³=(a+b)(a²−ab+b²),...; integrating 1/(ax²+bx+c) by completing the square; rationalizing substitutions (Example 9, Exercise 7.4/44) [pages 514-516 of Sec. 7.4]
• Lecture 32 (Fri, Apr 3): Strategies for integration: recalling the table of standard integrals and discussing possible strategies for integration [Sec. 7.5, in particular Exercises 7.5/19, 43, 48]
• Lecture 33 (Mon, Apr 6): Integration using tables and computer algebra systems (CASs): using tables of integrals; using the computer algebra system Mathematica (available freely for OU students) [Sec. 7.6; the Mathematica notebook and a pdf file with the same content from the lecture on April 6, illustrating the usage of Mathematica]
• Lecture 34 (Wed, Apr 8): Approximate integration: methods for approximate computation of definite integrals: left Riemann sums, right Riemann sums, midpoint rule, trapezoidal rule, Simpson's rule; idea of Simpson's rule - approximating the integrand locally by parabolas through three consecutive points; errors of the different methods - the errors of the left and right Riemann sum are of order (Δx)¹, the errors of the midpoint and the trapezoidal rules are of order (Δx)², the error of the Simpson's rule is of order (Δx)⁴; a handout with a list of the different methods and the behavior of their methods, and a numerical example [Sec. 7.7; the Mathematica notebook and a pdf file with the same content illustrating the different methods for approximate integration]
• Lecture 35 (Fri, Apr 10): Approximate integration (cont.): a recap of the five methods for approximate computation of definite integrals studied in Lecture 34 - a geometric interpretation and expression for the errors
  Improper integrals: two types of possible problems with integrals - infinite integration interval and discontinuous integrand; definition of an improper integral of type 1 as a limit of an integral over a finite interval [a,b] as a→−∞ and/or b→∞; examples [pages 543-547 of Sec. 7.8]
• Lecture 36 (Mon, Apr 13): Improper integrals (cont.): improper integrals of type 2 (whose integrand has an infinite discontinuity at an endpoint or at an internal point of the integration interval); definition of an improper integral of type 2 over [a,b] as an one-sided limit of an integral over a subinterval of [a,b] when the discontinuity of the integrand occurs at an endpoint;l definition of an improper integral of type 2 as the sum of two one-sided limits when the discontinuity of the integrand occurs inside (a,b); dealing with "doubly" improper integrals, i.e., integrals over infinite regions and with an integrand that has an infinite discontinuity - break the integral into a sum of an integral of type 1 and an integral of type 2 (by definition, the improper integral converges only if both integrals converge); a comparison test for improper integrals; examples [pages 547-550 of Sec. 7.8]
• Lecture 37 (Wed, Apr 15): Partial fractions / Aproximate integration (cont.): Mock Quiz 10 (with detailed solutions and additional questions).
  Improper integrals (cont.): an example of using comparison theorems: a proof that integral of x^−1/2e^−x from 0 to ∞ converges by writing it as a sum of an integral from 0 to 5 and an integral from 5 to ∞; on the interval [0,5], x^−1/2e^−x<x^−1/2, and integral of x^−1/2 from 0 to 5 is 2(5)^1/2; on the interval [5,∞), x^−1/2e^−x<5^−1/2e^−x, and integral of 5^−1/2e^−x from 5 to ∞ is 5^−1/2e⁻⁵; we can conclude that the original "doubly" improper integral converges and that its value is no greater than 2(5)^1/2+5^−1/2e⁻⁵=4.475149...; the exact value of the integral is π^1/2=1.77245385... [Sec. 7.8]
• Lecture 38 (Fri, Apr 17): Arc length: a review of using definite integrals to compute the volume of a solid of revolution by slicing it with planes perpendicular to the axis of revolution, and by the method of cylindrical shells; derivation of the expression of the arc lengh of a curve with equation y=f(x) (and, by analogy, of a curve with equation x=g(y)); arclength function s(x) [Sec. 8.1]
• Lecture 39 (Mon, Apr 20): Arc length (cont.): applying the formula for arc length to compute the length of a straight line between two points in the plane; the arclength function s(x) for the half circle of unit radius centered at the origin in the upper half-plane [Sec. 8.1]
  Area of a surface of revolution: derivation of an expression for the (side) area of a cone of base radius r and slant height l by unfolding it; derivation of the area A given by equation (2) on page 570 by ''moving an infinitesimally short segment of a straight line in a direction perpendicular to it'' - the swept area is approximately equal to the length traveled by the center point of the segment, multiplied by the length of the segment - apply this to derive equation (2) [pages 569, 570 of Sec. 8.2]
• Lecture 40 (Wed, Apr 22): Area of a surface of revolution (cont.): derivation of an expression for the (side) area of a surface of revolution obtained by rotating the curve y=ƒ(x) around the x-axis; computation of the volume of the Gabriel's horn (Exercise 8.2/25) by the method of slicing it by planes perpendicular to the x-axis, and by the method of cylindrical shells; demonstration that the (side) area of the Gabriel's horn is infinite by using the Comparison Theorem [Sec. 8.2]
• Lecture 41 (Fri, Apr 24): Exam 3 [on Sections 7.1-7.8, 8.1, covered in Lectures 25, 27-38]
• Lecture 42 (Mon, Apr 27): Modeling with differential equations: a digression on solving algebraic equations: proving the existence of a root of the algebraic equation ƒ(x)=0 in an interval [a,b] for a continous function ƒ by using the Intermediate Value Theorem, and computing the numerical value of the root by using the bisection method of Newton's method; an ordinary differential equation (ODE) of order n, initial condition(s) (ICs) for an order-n ODE (an ODE of order n needs n initial conditions); an example: y'(t)=7 (ODE), y(5)=19 (IC) - the general solution of the ODE is the family of functions y(t)=7t+C (where C is an arbitrary constant), and the particular solution to the ODE and the IC is the function y(t)=7t−16 from this family [pages 89, 90 of Sec. 1.8; pages 263-265 of Sec. 3.8; pages 606-608 of Sec. 9.1]
• Lecture 43 (Wed, Apr 29): A digression: a mathematical proof that if at 8:00 a.m. Elevator #1 in PHSC was at a lower floor than Elevator #2, and at 9:00 a.m. Elevator #1 was at a higher floor than Elevator #2, then there exists a moment of time between 8:00 a.m. and 9:00 a.m. at which both elevators were at the same height, by applying the Intermediate Value Theorem to the function h(t)=ƒ₂(t)−ƒ₁(t) where ƒ_j(t) is the height of Elevator #j at time t.
  Models for population growth: derivation of the law of natural growth, P'(t)=aP(t); derivation of the general solution P(t)=Ce^at of P'=aP, and of the particular solution P(t)=P₀e^at of P'=aP, P(0)=P₀; accounting for the finite resources - derivation of the logistic equation, P'(t)=aP(t)[1−P(t)/K], where K=const is the carrying capacity of the ecosystem; solving the logistic equation by using partial fraction decomposition [pages 629-633 of Sec. 9.4]
• Lecture 44 (Fri, May 1) Last lecture: finding the indefinite integral of (1−x)^1/2 by using:
  • trigonometric substitution x=sin t and the formulas cos²t = (1/2)(1+cos 2t) and sin 2t = 2 sin t cos t;
  • integration by parts; and
  • interpreting the definite integral of (1−x)^1/2 from 0 to a as the area under the graph of the function ƒ(x) = (1−x)^1/2 from x = 0 to x = a.
  Last words of wisdom and advice about math and life in general, tears, etc.
• Final Exam, May 6 (Wed), 1:30-3:30 p.m., 201 PHSC

Discussion sections