Math 1314 Lesson 7 Applications of the Derivative. rate of change, instantaneous rate of change, velocity, => the derivative

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1 Math 1314 Lesson 7 Applications of the Derivative In word problems, whenever it s anything about a: rate of change, instantaneous rate of change, velocity, => the derivative average rate of change, difference quotient =>an average *An interval MUST be given!!* A common use of rate of change is to describe the motion of an object. The function gives the position of the object with respect to time, so it is usually a function of t instead of x. If the object changes position over time, we can compute its rate of change, which we refer to as velocity. We can find either the average rate of change or the instantaneous rate of change, depending on the question posed. Velocity can be positive, negative or zero. If you throw a rock up in the air, its velocity will be positive while it is moving upward and will be negative while it is moving downward. We refer to the absolute value of velocity as speed. Velocity has two components: speed and direction. Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 1

2 Many of our problems will ask for the rate at which something is changing at a specific number. Other times, the problem will ask for a function value or an average rate of change. From there we ll apply the appropriate method. Example 1: Suppose the distance covered by a car can be measured by the function f () t 4t 3t, where f () t is given in feet and t is measured in seconds. a. Find the distance covered by the car in 10 seconds. b. Find the rate of change of the car when t 4. c. Find the average velocity of the car over the interval [1, 6]. f ( th) f( t) Recall:. h Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative

3 Example : A study conducted for a specific company showed that the number of lawn chairs assembled by the typical worker t hours after starting work at 6 a.m. is given by 3 Nt () t 7t 18t. a. At what rate will the typical worker be assembling lawn chairs at 9 a.m.? b. How many lawn chairs will the typical worker have assembled by 1 p.m.? c. What is the average rate at which the lawn chairs are assembled from 7 a.m. to 11 a.m. Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 3

4 Example 3: The median price of a home in one part of the US can be modeled by the function Pt ( ) t 9.637t 15.84, where P(t) is given in thousands of dollars and t is the number of years since the beginning of According to the model, at what rate were median home prices changing at the beginning of 005? Example 4: A country s gross domestic product (in millions of dollars) is modeled by the 3 function Gt ( ) t 45t 0t 6000 where 0 t 11 and t 0 corresponds to the beginning of What was the average rate of growth of the GDP over the period ? Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 4

5 Example 5: Find the slope of the tangent line of Enter the function into GGB. 3 ( ) x ln( ) f x x e x when x = 4. Example 6: Find the numerical derivative of Enter the function into GGB. x f( x) x x 3 when x = -1. We can also find the equation of the tangent line by using the following GGB command. tangent[<x-value>,<function>] 3 Example 7: Give the equation of the line tangent to f( x) 1.6x 6.39x.81 at (3, 59.56). Enter the function into GGB. Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 5

6 In other cases, we may want to find all values of x for which the tangent line to the graph of f is horizontal. Since the slope of any horizontal line is 0, we ll want to find the derivative, set it equal to zero and solve the resulting equation for x. Example 8: Find all x-values on the graph of f x 5 3x x horizontal. where the tangent line is We can also determine values of x for which the derivative is equal to a specified number. Set the derivative equal to the given number and solve for x either algebraically or by graphing Example 9: Find all values of x for which f '( x) 3 : f ( x) x x 7x 3 3 Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 6

7 Sometimes we need to find the derivative of the derivative. Since the derivative is a function, this is something we can readily do. The derivative of the derivative is called the second derivative, and is denoted f ''( x). To find the second derivative, we will apply whatever rule is appropriate given the first derivative. An application of the second derivative would be acceleration, as it s the rate at which velocity is changing. Example 10: Find the second derivative: f x x x x 5 ( ) Example 11: Find the value of the second derivative when x = 5 if Enter the function into GGB. x ln x f( x) ( x 3) 1 3. Lesson 7 Derivatives at a Point, Numerical Derivatives and Applications of the Derivative 7