Application of derivatives

18.01 Calculus Jason Starr Fall 2005 Lecture 1. September 8, 2005Homework. Problem Set 1 Part I: (a)–(e); Part II: Problems 1 and 2.Practice Problems. Course Reader: 1B11 1B­Textbook: 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, Δss(t0 + Δt) − s(t0) vave == . Δt Δt Velocity, or instantaneous velocity, at t0 is, v(t0 ) = lim vave = lim s(t0 + Δt) − s(t0) . Δt0Δt0 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) . Δt0 Δt→Example. For s(t) = −5t2 + 20t, first computed velocity at t = 1 is, v(1) = lim 10 − 5Δt = 10. Δt0→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). 18.01 Calculus Jason Starr Fall 2005 The difference quotient or average rateof­cchang of y from x0 to x0 + Δx is, Δyf (x0 + Δx) − f (x0) = . Δx Δx The derivative of y (or f (x)) with respect to x at x0 is, Δyf (x0 + Δx) − f (x0)lim = lim . Δx 0 Δx Δx0 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,1V = 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, 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 ofa cone.