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Chapter 3: Applications of Derivatives 3.1 Velocity Recall: s: position 𝑑𝑠 : velocity 𝑑𝑡 𝑑𝑣 𝑑𝑡 𝑑2 𝑠 = 𝑑𝑡 2 : acceleration (will come back to this) Position graphs: Step 1: Step 2: Step 3: Step 4: Find the initial and final position Find time when v = 0. Find position when v = 0. Sketch the position on the number line. Displacement: how far the object has travelled from the original Distance: how far the object has travelled in total 3.2 Acceleration If an object moves along a straight line, acceleration is the rate of change of velocity with respect to time. Therefore, it is the second derivative. Rates of Change in the Natural Sciences (3.3) Derivatives can be interpreted as rates of change. So now we’re going to use derivatives to find rates of change in physics, biology and chemistry. Physics Application average density is: ∆𝑚 𝑓(𝑥2 ) − 𝑓(𝑥1 ) = ∆𝑥 𝑥2 − 𝑥1 linear density at x will be the limit of the average density as ∆𝑥 approaches 0: 𝑝 = lim ∆𝑚 ∆𝑥→0 ∆𝑥 = 𝑑𝑚 𝑑𝑥 (just find the derivative) Biology Application If the number of items in a population at time t is n = f(t), then: ∆𝑛 𝑓(𝑡2 )−𝑓(𝑡1 ) Average rate of growth = Instantaneous rate of growth = lim ∆𝑡 = 𝑡2 −𝑡1 ∆𝑛 ∆𝑥→0 ∆𝑡 = 𝑑𝑛 𝑑𝑡 Chemistry Application The concentration of a substance A is the number of moles (6.022𝑥1023 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠) per litre and is denoted by [A]. During a chemical reaction the concentration will vary and so [A] is a function of time. During a time interval 𝑡1 ≤ 𝑡 ≤ 𝑡2 , the average rate of reaction of a reactant A is: [𝐴](𝑡2 ) − [𝐴](𝑡1 ) ∆[𝐴] =− ∆𝑡 𝑡2 − 𝑡1 and the instantaneous rate of reaction is the rate of change of concentration with respect to time. Rate of reaction= lim ∆[𝐴] ∆𝑡→0 ∆𝑡 =− 𝑑[𝐴] 𝑑𝑡 Since the rate of reaction is the derivative of the concentration function, chemists often determine the rate of change of reaction by measuring the slope of a tangent. Rates of Change in the Social Sciences (3.4) We will look at rates of change in business and economics. Cost Function: [C(x)] = the cost of producing x units of a certain commodity. Average rate of change in cost: ∆𝐶 𝐶(𝑥2 ) − 𝐶(𝑥1 ) 𝐶(𝑥1 + ∆𝑥) − 𝐶(𝑥1 ) = = ∆𝑥 𝑥2 − 𝑥1 ∆𝑥 Marginal Cost: [C’(x)] = the instantaneous rate of change of cost with respect to number of units produced. ∆𝐶 𝑑𝐶 = ∆𝑥→0 ∆𝑥 𝑑𝑥 lim We approximate by taking ∆𝑥 = 1, so that for large values of x, C’(x) is approximately equal to C(x + 1) – C(x). Demand (Price) Function [p(x)] = the price per unit that a company can charge if it sells x units. Revenue Function [R(x) = xp(x)] = total revenue of selling x units at a unit price of p(x). Marginal Revenue Function [R’(x)] = the rate of change in revenue with respect to the number of units sold. Profit Function [P(x) = R(x) – C(x)] = the difference between cost and revenue Marginal Profit Function number of units. [P’(x)] = the rate of change in profit with respect to the Profit will be maximized when first derivative = 0 means horizontal line, slope of 0 (maximum on a parabola) Related Rates (3.5) We are given the rate of change of one quantity and we are asked to find the rate of change of a related quantity. We find an equation that relates the 2 (or more) quantities and use the Chain Rule to differentiate BOTH sides with respect to time.