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18.01 Calculus Jason Starr Fall 2005 Math 18.01 Lecture Summaries Homework. These are the problems from the assigned Problem Set which can be completed using the material from that date’s lecture. Practice Problems. Practice problems are not to be written up or turned in. These are assigned only for practice, and are entirely voluntary. Problems listed as “1B-1”, for example, are taken from Section E of the 18.01 course reader. Lecture 1. Sept. 8 Velocity and derivatives Lecture 2. Sept. 9 Limits Lecture 3. Sept. 13 Rules of differentiation Lecture 4. Sept. 15 The chain rule and implicit differentiation Lecture 5. Sept. 16 The derivatives of exponential and logarithm functions Lecture 6. Sept. 20 The derivatives of trigonometric functions Lecture 7. Sept. 22 Review for Exam 1 Lecture 8. Sept. 27 Linear and quadratic approximations Lecture 9. Sept. 29 Sketching curves Lecture 10. Sept. 30 Applied maximum/minimum problems Lecture 11. Oct. 4 Related rates problems Lecture 12. Oct. 6 Newton’s method Lecture 13. Oct. 13 Antidifferentiation Lecture 14. Oct. 14 Riemann integrals Lecture 15. Oct. 18 The Fundamental Theorem of Calculus Lecture 16. Oct. 20 Properties of the Riemann integral Lecture 17. Oct. 21 Separable ordinary differential equations Lecture 18. Oct. 25 Numerical integration Lecture 19. Oct. 28 Applications of integration to volumes Lecture 20. Nov. 1 Averages and volumes by shells Lecture 21. Nov. 3 Parametric equation curves and arc length Lecture 22. Nov. 4 Area of a surface of revolution and polar coordinate curves Lecture 23. Nov. 8 Tangent lines, arc length and areas for polar curves Lecture 24. Nov. 15 Inverse trigonometric functions and hyp erbolic functions Lecture 25. Nov. 17 Inverse hyperbolic functions and inverse substitution Lecture 26. Nov. 18 Partial fraction decomposition Lecture 27. Nov. 22 Integration by parts 118.01 Calculus Jason Starr Fall 2005 Lecture 28. Dec. 1 L’Hospital’s rule Lecture 29. Dec. 2 Improper integrals Lecture 30. Dec. 6 Sequences and series Lecture 31. Dec. 8 Power series and Taylor series Lecture 32. Dec. 8 More Taylor series and review Lecture 1. September 8, 2005 Homework. Problem Set 1 Part I: (a)–(e); Part II: Problems 1 and 2. Practice Problems. Course Reader: 1B-1, 1B-2 Textbook: p. 68, Problems 1–7 and 15. 1. Velocity. Displacement is s(t). Increment from t0 to t0 + Δt is, Δs = s(t0 + Δt) − s(t0). Average velocity from t0 to t0 + Δt is, Δs s(t0 + Δt) − s(t0) vave = = . Δt Δt Velocity, or instantaneous velocity, at t0 is, v(t0 ) = lim vave = lim s(t0 + Δt) − s(t0) . Δt 0 Δt 0 Δt→ →This is a derivative, v(t) equals s�(t) = ds/dt. The derivative of velocity is acceleration, a(t0) = v�(t0 ) = lim v(t0 + Δt) − v(t0) . Δt 0 Δt→Example. For s(t) = −5t2 + 20t, first computed velocity at t = 1 is, v(1) = lim 10 − 5Δt = 10. Δt 0→Then computed velocity at t = t0 is, v(t0) = lim 0 −10t0 + 10 − 5Δt = −10t0 + 20. Δt→Finally, computed acceleration at t = t0 is, a(t0) = lim 0 −10 = −10. Δt→2. Derivative. Let y = f (x) be a dependent variable depending on an independent variable x, varying freely. The increment of y from x0 to x0 + Δx is, Δy = f (x0 + Δx) − f (x0). 218.01 Calculus Jason Starr Fall 2005 The difference quotient or average rate-of-change of y from x0 to x0 + Δx is, Δy f (x0 + Δx) − f (x0) = . Δx Δx The derivative of y (or f (x)) with respect to x at x0 is, Δy f (x0 + Δx) − f (x0)lim = lim . Δx 0 Δx Δx 0 Δx→ →3. Examples in science and math. (i) Economics. Marginal cost is the derivative of cost with respect to some other variable, for instance, the quantity purchased. (ii) Thermodynamics. The ideal gas law relating pressure p, volume V , and temperature T of a gas is, pV = nRT. Under isothermal conditions, T is a constant T0 so that, p(V ) = 0 V . nRTUnder adiabatic conditions (i.e., no transfer of heat), pV γ is a constant K. Using this to eliminate p gives, T (V ) = K nR 1 V γ−1 . As this illustrates, the independent variable, dependent variable and constants in an equation very much depend on the problem to be solved. (iii) Biology. Exponential population growth models the population N (t) after t years as, N (t) = N0e rt , where ex is the exponential function, N0 is initial population, and r is a growth factor. Later we will see, N �(t) = rN (t), i.e., the population grows at a rate proportional to the size of the population. (iv) Geometry. The volume of a right circular cone is, 1 V = A × h. 3 where A is the base area of the cone and h is the height of the cone. The radius r of the base is proportional to the height, r(h) = ch, 318.01 Calculus Jason Starr Fall 2005 for some constant c. Since A = πr2, this gives, π V (h) = c 2h3 . 3 The derivative is, dV = πc 2h2 = πr 2 = dh A. This is very reasonable. In some sense, this explains the classical formula for the volume of a cone. Lecture 2. September 9, 2005 Homework. Problem Set 1 Part I: (f)–(h); Part II: Problems 3. Practice Problems. Course Reader: 1C-2, 1C-3, 1C-4, 1D-3, 1D-5. 1. Tangent lines to graphs. For y = f(x), the equation of the secant line through (x0, f(x0)) and (x0 + Δx, f(x0 + Δx)) is, y = f(x0 + Δx) − f(x0)(x − x0) + f(x0). Δx In the limit, the equation of the tangent line through (x0, f(x0)) is, y = f�(x0)(x − x0) + y0. Example. For the parabola y = x2, the derivative is, y�(x0) = 2x0. The equation of the tangent line is, y = 2x0(x − x0) = 2x0x − x2 0. For instance, the equation of the tangent line through (2, 4) is, y = 4x − 4. 2Given a point (x, y), what are all points (x0, x0) on the parabola whose tangent line contains (x, y)? To solve, consider x and y as constants and solve for x0. For instance, if (x, y) = (1, −3), this gives, 2(−3) = 2x0(1) − x0, or, 2 x0 − 2x0 − 3 = 0. 418.01 Calculus Jason Starr Fall 2005 Factoring (x0 − 3)(x0 + 1), the solutions are x0 equals −1 and x0 equals 3. The corresponding tangent lines are, y = −2x − 1, and y = 6x − 9. For general (x, y), the solutions are, x0 = x ± � x2 − y. 2. Limits. Precise definition is on p. 791 of Appendix A.2. Intuitive definition: limxf (x)→x0 equals L if and only if all values of f (x) can be made arbitrarily close to L by
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