18.01 Calculus Jason Starr Fall 2005 Lecture 1. September 8, 2005 Homework. Problem Set 1 Part I: (a)–(e); Part II: Problems 1 and 2. Practice Problems. Course Reader: 1B1, 1B2 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, vave = Δs s(t0 + Δt) − s(t0 ) = . Δt Δt Velocity, or instantaneous velocity, at t0 is, s(t0 + Δt) − s(t0 ) . Δt→0 Δt v(t0 ) = lim vave = lim Δt→0 This is a derivative, v(t) equals s� (t) = ds/dt. The derivative of velocity is acceleration, v(t0 + Δt) − v(t0 ) . Δt→0 Δt a(t0 ) = v � (t0 ) = lim 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 −10t0 + 10 − 5Δt = −10t0 + 20. Δt→0 Finally, computed acceleration at t = t0 is, a(t0 ) = lim −10 = −10. Δt→0 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 ). 18.01 Calculus Jason Starr Fall 2005 The diﬀerence quotient or average rateofchange 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 . Δx→0 Δx→0 Δx Δx lim 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 ) = nRT0 . V Under adiabatic conditions (i.e., no transfer of heat), pV γ is a constant K. Using this to eliminate p gives, K 1 T (V ) = nR . 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) = N0 ert , 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, 18.01 Calculus Jason Starr Fall 2005 for some constant c. Since A = πr2 , this gives, V (h) = π 2 3 ch. 3 The derivative is, dV = πc2 h2 = πr2 = A. dh This is very reasonable. In some sense, this explains the classical formula for the volume of a cone.