Calculus Applications of Differentiations (IV)
Outline 1 Rates of Change In Economics And The Sciences Applications of Derivative In Economics Applications of Derivative in the Sciences 2 Related Rate Parametric Equations (I)
Outline 1 Rates of Change In Economics And The Sciences Applications of Derivative In Economics Applications of Derivative in the Sciences 2 Related Rate Parametric Equations (I)
Overview In this section, we round out our exposition of the derivative by presenting a collection of applications selected from a variety of fields. The applications in this section represent but a small sampling of some elementary uses of the derivative. These are not all of the uses of the derivative nor are they necessarily the most important uses. Our intent is simply to present some interesting applications in a variety of settings.
Marginal Cost and Marginal Profit (I) Recall that the derivative of a function gives the instantaneous rate of change of that function. So, when you see the word rate, you should be thinking derivative, for instance, inflation rate interest rate inflation rate These can be thought of as derivatives.
Marginal Cost and Marginal Profit (II) There are also many quantities with which you are familiar, but that you might not recognize as rates of change. Our first example, which comes from economics, is of this type. In economics, the term marginal is used to indicate a rate. Thus, marginal cost is the derivative of the cost function, marginal profit is the derivative of the profit function and so on. We introduce marginal cost in some detail here, with further applications given in the exercises.
Marginal Cost and Marginal Profit (III) Suppose that you are manufacturing an item, where your start-up costs are $4000 and productions costs are $2 per item. If you make x items, then your total cost would be 4000 + 2x. Of course, the assumption that the cost per item is constant is unrealistic. Efficient mass production techniques could reduce the cost per item, but machine maintenance, labor, plant expansion and other factors could drive costs up as production (x) increases.
Marginal Cost and Marginal Profit (IV) In the following example, a quadratic cost function is used to take into account some of these extra factors. In practice, you would find a cost function by making some observations of the cost of producing a number of different quantities and then fitting that data to the graph of a known function. (This is one way in which the calculus is brought to bear on real-world problems.) When the cost per item is not constant, an important question for managers to answer is how much it will cost to raise production. This is the idea behind marginal cost.
Analyzing the Marginal Cost of Producing a Commercial Product Example (7.1) Suppose that C(x) = 0.02x 2 + 2x + 4000 is the total cost (in dollars) for a company to produce x units of a certain product. Compute the marginal cost at x = 100 and compare this to the actual cost of producing the 100th unit.
Average Cost Another quantity that businesses use to analyze production is average cost. You can easily remember the formula for average cost by thinking of an example. If it costs a total of $120 to produce 12 items, then the average cost would be $10 ( ) 120 12 per item. In general, the total cost is given by C(x) and the number of items by x, so average cost is defined by C(x) = C(x) x. A business manager would want to know the level of production that minimizes average cost.
Minimizing the Average Cost of Producing a Commercial Product (I) Example (7.2) Suppose that C(x) = 0.02x 2 + 2x + 4000 is the total cost (in dollars) for a company to produce x units of a certain product. Find the production level x that minimizes average cost.
Minimizing the Average Cost of Producing a Commercial Product (II) Figure: [3.86] Average cost function.
Elasticity of Demand (I) Our third example also comes from economics. This time, we will explore the relationship between price and demand. Clearly, in most cases a higher price will lower the demand for the product. However, if sales do not decrease significantly, the company may actually bring in more revenue due to the price increase. As we will see, an analysis of the elasticity of demand can give us important information about revenue.
Elasticity of Demand (II) Suppose that the demand x for an item is a function of its price p. That is, x = f (p). If the price changes by a small amount p, then the relative change in price equals p. However, the p change in price would create a change in demand x, with a relative change in demand of x. Economists define the x elasticity of demand at price p to equal the relative change in demand divided by the relative change in price for very small changes in price. As calculus students, you can define the elasticity E as a limit: E = lim p 0 x x p p.
Elasticity of Demand (III) In the case where x is a function of p, we write p = (p + h) p = h for some small h and then x = f (p + h) f (p). We then have E = lim h 0 f (p+h) f (p) f (p) h p = p f (p) lim f (p + h) f (p) = p h 0 h f (p) f (p), assuming that f is differentiable. In example 7.3, we analyze elasticity of demand and revenue. Recall that if x = f (p) items are sold at price p, then the revenue equals pf (p).
Computing Elasticity of Demand and Changes in Revenue (I) Example (7.3) Suppose that f (p) = 400(20 p) is the demand for an item at price p (in dollars) with p < 20. 1 Find the elasticity of demand. 2 Find the range of prices for which E < 1. Compare this to the range of prices for which revenue is a decreasing function of p.
Computing Elasticity of Demand and Changes in Revenue (II) Figure: [3.87] E = p p 20.
Outline 1 Rates of Change In Economics And The Sciences Applications of Derivative In Economics Applications of Derivative in the Sciences 2 Related Rate Parametric Equations (I)
Rate of a Chemical Reaction (I) The next example we offer comes from chemistry. It is very important for chemists to have a handle on the rate at which a given reaction proceeds. Reaction rates give chemists information about the nature of the chemical bonds being formed and broken, as well as information about the type and quantity of product to expect. A simple situation is depicted in the schematic A + B C, which indicates that chemicals A and B (the reactants) combine to form chemical C (the product).
Rate of a Chemical Reaction (II) Let [C](t) denote the concentration (in moles per liter) of the product. The average reaction rate between times t 1 and t 2 is [C](t 2 ) [C](t 1 ) t 2 t 1. The instantaneous reaction rate at any given time t 1 is then given by [C](t) [C](t 1 ) lim t t 1 t t 1 = d[c] (t 1 ). dt Depending on the details of the reaction, it is often possible to write down an equation relating the reaction rate d[c] to the dt concentrations of the reactants, [A] and [B].
Modeling the Rate of a Chemical Reaction (I) Example (7.4) In an autocatalytic chemical reaction, the reactant and the product are the same. The reaction continues until some saturation level is reached. From experimental evidence, chemists know that the reaction rate is jointly proportional to the amount of the product present and the difference between the saturation level and the amount of the product. If the initial concentration of the chemical is 0 and the saturation level is 1 (corresponding to 100%), then the concentration x(t) of the chemical satisfies the equation x (t) = rx(t)[1 x(t)], where r > 0 is a constant. Find the concentration of chemical for which the reaction rate x (t) is a maximum.
Modeling the Rate of a Chemical Reaction (II) Figure: [3.88] y = rx(1 x).
Analyzing a Titration Curve (I) Our second example from chemistry involves the titration of a weak acid and a strong base. In this type of titration, a strong base is slowly added to a weak acid. The ph of the mixture is monitored by observing the color of some ph indicator, which changes dramatically at what is called the equivalence point. The equivalence point is then typically used to compute the concentration of the base.
Analyzing a Titration Curve (II) A generalized titration curve is shown in Figure 3.89, where the horizontal axis indicates the amount of base added to the mixture and the vertical axis shows the ph of the mixture. Notice the nearly vertical rise of the graph at the equivalence point. Figure: [3.89] Acid titration.
Analyzing a Titration Curve (III) Let x be the fraction (0 < x < 1) of base added (equal to the fraction of converted acid; see Harris Quantitative Chemical Analysis for more details), with x = 1 representing the equivalence point. Then the ph is approximated by c + ln x 1 x, where c is a constant closely related to the acid dissociation constant.
Example (7.5) Analyzing a Titration Curve (IV) Find the value of x at which the rate of change of ph is the smallest. Identify the corresponding point on the titration curve in Figure 3.89. Figure: [3.89] Acid titration.
Mass Density (I) Calculus and elementary physics are quite closely connected historically. It should come as no surprise, then, that physics provides us with such a large number of important applications of the calculus. We have already explored the concepts of velocity and acceleration. Another important application in physics where the derivative plays a role involves density. There are many different kinds of densities that we could consider. For example, we could study population density (number of people per unit area) or color density (depth of color per unit area) used in the study of radiographs. However, the most common type of density discussed is mass density (mass per unit volume).
Mass Density (II) You probably already have some idea of what we mean by mass density, but how would you define it? If the object of interest is made of some homogeneous material (i.e., the mass of any portion of the object of a given volume is the same), then the mass density is simply mass density = mass volume and this quantity is constant throughout the object. However, if the mass of a given volume varies in different parts of the object, then this formula only represents the average density of the object. In the next example we find a means of computing the mass density at a specific point in a nonhomogenous object.
Mass Density (III) Suppose that the function f (x) gives us the mass (in kilograms) of the first x meters of a thin rod (see Figure 3.90). The total mass between marks x and x 1 (x > x 1 ) is given by f (x) f (x 1 ) kg. The average linear density (i.e., mass per unit length) between x and x 1 is then defined as f (x) f (x 1 ) x x 1. Figure: [3.90] A thin rod.
Mass Density (IV) Finally, the linear density at x = x 1 is defined as ρ(x 1 ) = lim x x1 f (x) f (x 1 ) x x 1 = f (x 1 ), where we have recognized the alternative definition of derivative discussed in section 2.2. Figure: [3.90] A thin rod.
Density of a Thin Rod Example (7.6) Suppose that the mass in a thin rod is give by f (x) = 2x. Compute the linear density at x = 2 and at x = 8 and compare the densities at the two points.
Modeling Electrical Current in a Wire (I) The next example also comes from physics, in particular from the study of electromagnetism. Suppose that Q(t) represents the electrical charge in a wire at time t. Then, the derivative Q (t) gives the current flowing through the wire. To see this, consider the cross section of a wire as shown in Figure 3.91. Figure: [3.91] An electrical wire.
Modeling Electrical Current in a Wire (II) Example (7.7) The electrical circuit shown in Figure includes a 5-ohm resistor, a 12-henry inductor, a 10-farad capacitor and a battery supplying 8 volts of AC current modeled by the oscillating function 8 sin 2t, where t is measured in seconds. Find the current in the circuit at any time t. Figure: [3.92] A simple electrical circuit.
Population Growth (I) In section 2.1, we briefly explored the rate of growth of a population. Population dynamics is an area of biology that makes extensive use of calculus. We examine population models in some detail in sections 6.1 and 6.2. For now, we explore one aspect of a basic model of population growth called the logistic equation.
Population Growth (II) This states that if p(t) represents population (measured as a fraction of the maximum sustainable population), then the rate of change of the population satisfies the equation p (t) = rp(t)[1 p(t)], for some constant r. A typical solution [for r = 1 and p(0) = 0.1] is shown in Figure 3.93. Figure: [3.93] Logistic growth.
Finding the Maximum Rate of Population Growth Example (7.8) Suppose that a population grows according to p (t) = 2p(t)[1 p(t)] (the logistic equation with r = 2). Find the population for which the population growth rate is a maximum. Interpret this point graphically.
Concluding Remarks (I) Notice the similarities between examples 7.4 and 7.8. One reason that mathematics has such great value is that seemingly unrelated physical processes often have the same mathematical description. Comparing examples 7.4 and 7.8, we learn that the underlying mechanisms for autocatalytic reactions and population growth are identical.
Concluding Remarks (II) We have now discussed examples of eight rates of change drawn from economics and the sciences. Add these to the examples we have seen in previous sections (velocity, van der Waal s equation etc.) and we have an impressive list of applications of the derivative. Even so, we have barely begun to scratch the surface. In any field where it is possible to quantify and analyze the properties of a function, calculus and the derivative are powerful tools.
Concluding Remarks (III) This list includes at least some portion of every major field of study. The continued study of calculus will give you the ability to read (and understand) technical studies in a wide variety of fields and to see (as we have in this section) the underlying unity that mathematics brings to human endeavors.
Outline 1 Rates of Change In Economics And The Sciences Applications of Derivative In Economics Applications of Derivative in the Sciences 2 Related Rate Parametric Equations (I)
Related Rate The first several examples in this section are known as related rates problems. The common thread in each problem is an equation relating two or more quantities that are all changing with time. In each case, we will use the chain rule to find derivatives of all terms in the equation (much as we did in section 2.8 with implicit differentiation). The differentiated equation allows us to determine how different derivatives (rates) are related.
A Related Rates Problem Example (8.1) An oil tanker has an accident and oil pours out at the rate of 150 gallons per minute. Suppose that the oil spreads onto the water in a circle at a 1 thickness of (see Figure 10 3.94). Given that 1 ft 3 equals 7.5 gallons, determine the rate at which the radius of the spill is increasing when the radius reaches 500 feet. Figure: [3.94] Oil spill.
A Guide Line for Solving Related Rate Problems Although the details change from problems to problems, the general pattern of solution is the same for all related rates problems. Looking back, you should be able to identify each of the following steps in example 8.1. 1 Make a simple sketch, if appropriate. 2 Set up an equation relating all of the relevant quantities. 3 Differentiate both sides of the equation with respect to time (t). 4 Substitute in values for all known quantities and derivatives. 5 Solve for the remaining rate.
Another Related Rates Problem Example (8.2) A car is traveling at 50 mph due south at a point 1 2 mile north of an intersection. A police car is traveling at 40 mph due west at a point 1 4 mile due east of the same intersection. At that instant, the radar in the police car measures the rate at which the distance between the two cars is changing. What does the radar gun register? Figure: [3.95] Car approaching an intersection.
Estimating a Rate of Change in Economics Example (8.3) A small company estimates that when it spends x thousand dollars for advertising in a year, its annual sales will be described by s(x) = 60 40e 0.05x thousand dollars. The four most recent annual advertising totals are give in the following table. Year 1 2 3 4 Dollars 14,500 16,000 18,000 20,000 Estimate the current (year 4) value of x (t) and the current rate of change of sales.
Tracking a Fast Jet Example (8.4) A spectator at an air show is trying to follow the flight of a jet. The jet follows a straight path in front of the observer at 540 mph. At its closest approach, the jet passes 600 feet in front of the person. Find the maximum rate of change of the angle between the spectator s line of sight and a line perpendicular to the flight path, as the jet flies by. Figure: [3.96] Path of jet.
Outline 1 Rates of Change In Economics And The Sciences Applications of Derivative In Economics Applications of Derivative in the Sciences 2 Related Rate Parametric Equations (I)
Parametric Equations for Circular Motion Example (8.5) The position of an object moving along a circle of radius 4 is described by the parametric equations x(t) = 4 cos 2t and y(t) = 4 sin 2t, where t is measured in seconds and x and y are measured in feet. Find the position, velocity (state the direction of motion) and speed at times 1 t = 0, 2 t = π 8, 3 t = π 4.
Parametric Equations for Circular Motion (II) Figure: [3.97] Path of object.
Velocity of the Scrambler (I) Example (8.6) For the path of the Scrambler x = 2 cos t + sin 2t, y = 2 sin t + cos 2t, find the horizontal and vertical components of velocity and speed at times t = 0 and t = π 2 and indicate the direction of motion. Also determine all times at which the speed is zero.
Velocity of the Scrambler (II) Figure: [3.98a] The Scrambler. Figure: [3.98b] Path of a Scrambler rider.