Section 3.4 The Derivative as a Rate of Change: Derivative measures the rate of change of a dependent variable to independent variable. If s=f(t) is the function of an object showing the position of the object as s at time t. Then Displacement= s = f(t + s) f(t). Average velocity = displacement travel time = s = f(t+ t) f(t) t t Velocity= ds = lim dt t 0 f(t+ t) f(t) t Acceleration= derivative of velocity Example 1 Particle Motion: At time t 0, the velocity of a body moving along the horizontal s-axis is v = t 2 4t + 3. (a) Find the body s acceleration each time the velocity is zero. (b) When is the body moving forward? Backward? (c) When is the body s velocity increasing? Decreasing? Example 2 Marginal revenue: Suppose that the revenue from selling x washing machines is r(x) = 20000(1 1 x ) dollars. 1
(a) Marginal revenue is defined to be the rate of change of revenue with respect to level production. Find the marginal revenue when 100 machines are produced. (b) Use the function r (x) to estimate the increase in revenue that will result from increasing production from 100,achines a week to 101 machines a week. (c) Find the limit of r (x) as x. How would you interpret this number? Example 3 When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10 6 + 10 4 t 10 3 t 2. Find the growth rates at (a) t=0 hours (b) t=5 hours (c) t=10 hours 2
Example 4 The Volume V = 4 3 πr3 of a spherical balloon changes with the radius. (a) At what rate (ft 3 /ft)does the volume change with respect to the radius when r=2 ft? (b) By approximately how much does the volume increase when the radius changes from 2 to 2.2? Section 3.8 Related Rates Idea: Compute the rate of change of one quantity in terms of the rate of change of another quantity. We need to relate two quantities by an equation. Given a problem about related rates: 1. Name all quantities 2. Relate quantities to each other by equations 3. Use chain rule to see the relation between rate of changes 4. Substitute the information given in the equations to solve for the unkowns Example 5 If y = x 3 + 2x and dx dt = 5, find dy dt when x=2. 3
Example 6 A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m 3 /min. How fast is the height of the water increasing? Example 7 A television camera is positioned 4000ft from the base of a rocket launching pad. the angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Lets assume the rocket rises vertically and its speed is 600 ft/s when it has risen 3000ft. (a) How fast is the distance from the television camera to the rocket changing at that moment? (b) If the television camera is always kept aimed at the rocket, how fast is the camera s angle of elevation changing at that same moment? 4
Section 3.9 Linearization and Differentials Purpose: Estimate functions by linear functions Recall the tangent line of a curve Tangent line equation at x=a: y-f(a)=f (a)(x-a) y = f(a) + f (a)(x a) f(x) f(a) + f (a)(x a) This is called linear approximation, tangent line approximation or Taylor polynomial of degree 1 of f(x) at x=a. Example 8 Approximate 4.05 Hint: Use f(x) = x Example 9 Use linear approximation to estimate tan (44) 5
Example 10 Find linearization of f(x) = x 2 + 9 at x = 4 Example 11 Find linearization of f(x) = sin (x) at x = 0 Exercise 12 Find linearization of f(x) = cos (x) and g(x) = tan (x) at x = 0. For y=f(x), we denote the derivative of y=f(x) as dy dx = f (x). dy and dx are called differentials, and dx = x = change in x, dy = f (x)dx = approximate change in y. Example 13 Find dy for 1. y = sin (x) 2. y = x2 x+1 3. xy 2 4x 3 2 y = 0 6
Example 14 Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05cm thick to a hemispherical dome with diameter 50m Hint: Volume of a sphere is 4 3 πr3 Example 15 The amount of work done by the heart s main pumping chamber, the left ventricle, is given by the equation W = P V + V δv2 2g where W is the work per unit time, P is the average blood pressure, V is the volume of blood pumped out during the unit of time, δ is the weight density of the blood, v is the average velocity of the exiting blood, and g is the acceleration of gravity. When P,V,δ and v remain constant, W becomes a function of g, and the equation takes the simplified form W = a + b g ( a, b constant ) As a member of NASA s medical team, you want to know how sensitive W is to apparent changes in g caused by flight maneuvers, and this depends on the initial value of g. As part of your investigation, you decide to compare the effect on W of a given change dg on the moon, where g = 5.2ft/sec 2, with the effect thhe same change dg would have on Earth, where g = 32ft/sec 2. Use the simplified equation above to find the ratio of dw moon to dw earth. 7