3.8 Exponential Growth and Decay

Transcription

1 3.8 Exponential Growth and Decay Suppose the rate of change of y with respect to t is proportional to y itself. So there is some constant k such that dy dt = ky The only solution to this equation is an exponential function of the form y(t) = y 0 e kt where y(0) = y 0. Verify this: Example: A bacteria culture starts with 5000 bacteria and triples every 2 hours. Assume the population grows at a rate proportional to its size. (a) Find a function that models how much bacteria is present after t hours. (b) When will the culture have 20,000 bacteria? (c) At what rate is the bacteria population growing after 30 minutes? 1

2 The population of a city grows at a rate proportional to its size. After 4 years, the city has 2,000 residents. After 6 years, the city has 10,000 residents. (a) Find the initial population. (b) When will the population be 18,000? 2

3 The half-life of a radioactive substance is the amount of time it takes for half of the substance to decay. This means that the rate of decay is proportional to the amount present at that moment. Example: Polonium-210 has a half-life of 140 days. If a sample has a mass of 200 mg, find the mass after t days. How long until the sample has only 40 mg? Example: Suppose that a radioactive substance decays to 40% of its original amount in 12 days. What is the half life of this substance? 3

4 Newton s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings. dy dt = k(y T ) where y is the temperature of the object and T is the room temperature (i.e. surroundings). The solution to this equation is a function of the form: the temperature of the where y 0 is the initial temperature of the object. y(t) = (y 0 T )e kt + T Example: An object with a temperature of 190 F is taken out of an oven and into a room with temperature 72 F. After 10 minutes the object has cooled to 175 F. When will the object be at 100 F? 4

5 3.9 Related Rates Related rate word problems deal with finding the rate of change (derivative) of one quantity in terms of the rate of change of some other related quantity. The goal is to create an equation that relates the quantities involved in the problem. Then, differentiate this equation with respect to time to get the related rate equation. 1. A circular area of snow is melting so that the radius is decreasing at a rate of 3 inches/day. At what rate is the area of the circle changing when the radius of the circle is 12 inches? 2. A 10-ft ladder rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 ft/s, how fast is the angle between the top of the ladder and the wall changing when the angle is π 4 radians? At what rate is the angle changing when the base of the ladder is 7 ft from the wall? 5

6 3. A baseball diamond is a square with 90 ft sides. A batter hits the ball and runs toward first base with a speed of 8 ft/s. (a) At what rate is his distance from third base increasing when he is halfway to first base? (b) Suppose at the same time the batter begins to run for first base, a man on third base begins to run for home plate at a speed of 7 ft/s. At what rate is the distance between the two men changing 4 seconds later? 6

7 4. A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, at what rate is his shadow on the building changing when he is 4 m from the building? 5. You are driving west on a road at 60 mph. A plane passes over you at an altitude of 5 miles flying east with a speed of 300 mph. At what rate is the distance between your car and the plane changing when the distance between you is 35 miles? 7

8 6. Water is being pumped into an inverted conical tank with radius 4 m and height 10 m at a rate of 5 m 3 /min. At what rate is the water level rising when the water has a height of 3 m? 8

9 7. A trough is 20 ft long and has ends which are isosceles triangles that are 4 ft across the top with a height of 5 ft. Water is being pumped in at some constant rate. If the water level is rising at a rate of 1 ft/hr when the height of the water is 2 ft, find the rate at which water is being pumped in. 9

10 3.10 Linear Approximations and Differentials The equation of the tangent line to the graph of f(x) at the point (a, f(a)) can be used to approximate values of f that are near a. This is called the linear approximation or tangent line approximation of f at a and the function L(x) = f(a) + f (a)(x a) is called the linearization of f at a. Example: Find the linearization of the function f(x) = sin x at x = π 3. Example: Find the linearization L(x) of the function f(x) = x at a = 25 and use it to approximate

11 Example: Use a linear approximation to find an approximate value for (15.8) 3/4. Example: For a given function, we know that f(7) = 4 and that f (x) = approximation to estimate f(7.05). 1 x 2 40 for all x. Use a linear Example: Suppose for a function f(x) that the linear approximation at a = 5 is given by the tangent line y = 2x 1. (a) What are f(5) and f (5)? (b) If g(x) = 1 f(x), find the linear approximation for g(x) at a = 5. 11

12 Differentials: We have seen the notation dy dx, which denotes the derivative of y with respect to x and gives the slope of the tangent line at x. Now, we will actually view dy and dx as separate entities, called differentials. Definition: The differential dy is defined by dy = f (x) dx. dy is an approximation for the actual change in y, which we denote y. Example: Find the differential dy of the function f(x) = (x 3 +x) 4 and then evaluate dy for x = 1, dx = Example: Compare the values of y and dy if f(x) = x 3 2x 2 + 3x 1 and x changes from 2 to Example: The side length of a cube was measured to be 10 cm with a maximum error in measurement of 0.2 cm. Use differentials to estimate the maximum possible error in the calculated surface area and volume of the cube. What are the relative errors and percentage errors for both? 12