MATH 234, Calculus for BusinessLecture 8, Textbook Sections 4.1More Examples of DerivativesFebruary 13th, 2017Announcements• There are many different notations for derivatives; be careful.• Today is the last day to be added to the list for the conflict exam.• Office hours are changed for this week only: Monday, 2/13, 11am to 12pm and also 3pm to 4pmin 24 Illini Hall. No office hours on Wednesday, 2/15.• Tutoring hours are changed for this week only (all in 441 Altgeld Hall):Day Time Teaching AssistantsMonday 4:00 to 6:00pm William Karr, Dara Zirlin6:00pm to 7:00pm Christopher Gartland7:00pm to 9:00pm Mingyu ZhaoTuesday 4:00 to 6:00pm Dileep Menon, Paulina Koutsaki6:00 to 8:00pm Artur Kirkoryan, Alessandro Gondoloand Lutian Zhao• Wednesday and Thursday tutoring hours are cancelled.• Thursday section is cancelled, but Tuesday section is still ON.ReviewDefinition. The instantaneous rate of change of f(x) at x = a isThis is also called the derivative of f(x) at x = a.Definition. The tangent line to the graph of y = f(x) at the point (a, f(a))is the .The equation of this tangent line is1Section 4.1, Techniques for Finding DerivativesDefinition (Differentiation, Version 1: Lagrange, 1793). The derivative of f (x) is the functionDefinition (Differentiation, Version 2: Leibniz, 1675). The derivative of f (x) is the functionDefinition (Differentiation, Version 3: Newton, 1666). The derivative of f (x) is the functionNote: The derivative inputs a function and outputs another function. Even more (10+) notations exist;people argue notations to this day.Example 1. Let f(x) = 2 for all x. Computedfdx(x).Solution.Theorem 1. Let k be a constant, real number and let f (x) = k for all x. Then f0(x) = .Similarly to before, the proof isExample 2. Let f(x) = 3x + 5. Compute f0(x) using the definition of the derivative.Solution.2Theorem 2. Let f (x) = mx + b (m, b are constant). Then f0(x) = .Similar to before, the proof isExample 3. Let f0(1) = 2 and g0(1) = 3. Define T (x) = 4f (x) + 5g(x). Compute T0(1) using the limit lawsand definition of the derivative.Solution.Theorem 3. Let f (x) and g(x) be differentiable functions (i.e., their derivatives exists).If T (x) = af(x) + bg(x), then T0(x) =Similarly to before, the proof is3Theorem 4 (The Sum Rule). Let f (x) and g(x) be differentiable functions (i.e., their derivatives exists).If T (x) = f(x) + g(x), then T0(x) =Theorem 5 (The Difference Rule). Let f(x) and g(x) be differentiable functions (i.e., their derivativesexists).If T (x) = f(x) − g(x), then T0(x) =Theorem 6 (The Constant Multiplication Rule). Let f (x) be a differentiable function and let k be constant.If T (x) = kf (x), then T0(x) = .Example 4. The parabola corresponding to a mystery quadratic function f (x) looks likeWhich of the four is the graph of the derivative f0(x)?(A) (B)(C) (D)Theorem 7 (As Derivative = Instantaneous Rate of Change...).(1) If f0(a) > 0, then f (x) is increasing at x = a (and vice versa)(2) If f0(a) < 0, then f (x) is decreasing at x = a (and vice versa).Solution to Example 4.4Example 5. Let f(x) = ax2+bx+c (a, b, c are constant). Compute f0(x) using the definition of the derivative.Solution.Example 6. The mystery quadratic function from Slide 12 is f (x) = x2− 2xUsing the previous slide, then f0(x) = .Note that f0(x) and f0(x) .As noted before, f(x) and f (x) .Example 7. Let f(x) = x3. Compute f0(x) using the definition of the derivative.Solution.5Theorem 8 (The Power Rule). If f (x) = xn, then f0(x) = . For example,f(x) f0(x)x2x3x4x5The proof is omitted (look up “Binomial Theorem”).Remark. The power rule is true even is n is NOT an integer. If f (x) =√x = x1/2,Example 8. Use the power rule and other properties of derivatives to computeddx4x16+ 3x5+ 7x + 1Solution.Definition. Let C(x) be the cost of producing x-units of a product. The marginal cost is .Definition. Let R(x) be the revenue of selling x-units of a product. The marginal revenue is .Definition. Let P (x) be the profit of producing and selling x-units of a product.The marginal profit is .Example 9. The cost for producing x-units of a product is C(x) = 3x2+ 200x + 300.(1) Find the marginal cost at x = 5 using power rule, etc.(2) What is the actual change in cost from x = 5 to x =
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