Section. - Derivatives and Rates of Change Recall : The average rate of change can be viewed as the slope of the secant line between two points on a curve. In Section.1, we numerically estimated the slope of a tangent line (instantaneous rate of change) by calculating the slopes of secant lines nearer and nearer to the point of tangency and determined what value they approached. In actuality, we were numerically taking the limit of the slopes of the secant lines. Definition: The derivative of a function f at a number a, denoted by f (a), is provided that this limit eists. Interpretations of the Derivative The derivative has various applications and interpretations, including the following: 1. Slope of the tangent line: f (a) is the slope of the line tangent to the graph of f at the point(a, f(a)).. Instantaneous rate of change: f (a) is the instantaneous rate of change of y= f() at =a. 3. Velocity: If f() is the position of a moving object at time, then v= f (a) is the velocity of the object at time =a. Eample 1: Given f()= + 8, find the following: a) f (). b) The equation of the tangent line at = 1
Eample : Given f()=, find the following: a) f (a). b) The equation of the tangent line at =9 Eample 3: The position of a particle is given by the equation of motion s(t) =, where t is measured in 1+t seconds and s is in meters. Find the velocity and the speed after seconds.
Eample : The temperature, T, in degrees Fahrenheit of a cold ham placed in a hot oven is given by T = f(t), where t is time in minutes since the ham was put in the oven. a) What is the meaning of f(0) = 75? b) What is the sign of f (t)? Why? c) What is the meaning and what are the units of f (0)=? Eample 5: The profit (in dollars) from the sale of car seats for infants is given by a) Find P(1000) and interpret. P()=5 0.05 5000,0 00 b) Find P (1000) and interpret. Section. Highly Suggested Homework Problems: 1, 5, 7, 11, 15, 17, 19, 3, 9, 31, 37, 39, 7 3
Section.7 - The Derivative as a Function Recall: The derivative of a function f at a specific number a is given by If the number a is allowed to vary, we obtain a new function called the derivative of f, f (), which is defined as Other Notations for the Derivative of y= f(): f ()=y = dy d = d f d = d d f() Eample 1: Given the graph of f() below, estimate the value of f () at the following points: y f() 5 3 1 3 1 1 8 7 5 3 1 1 3 5 7 8 9 1 3 5 f () Conclusion:
Eample : Find f () if f()= 1 +. State the domain of f() and f (). Noneistence of the Derivative If f (a) does not eist, then we say that f() is nondifferentiable at =a. This occurs when the graph: 1.. 3. 5
Eample 3: For what values of is f nondifferentiable? y 8 f() 8 8 8 Eample : Given the graph of f() below sketch the graph of f (). y 3 f() 1 3 1 1 3 1 3 y 3 1 3 1 1 3 1 Eample 5: Given the graph of f() below sketch the graph of f (). y 10 y 10 3 8 f() 8 10 8 8 10 10 8 8 10 8 8 10 10
Definition: If f is a differentiable function, then its derivative f is also a function, so f may have a derivative of its own, denoted by ( f ) = f. This new function f is called the second derivative of f because it is the derivative of the derivative of f. f ()=y = d ( ) dy = d y d d d Position, Velocity, and Acceleration The position function of an object that moves in a straight line is usually given by s(t). The velocity of an object as a function of time is given by The acceleration of an object is the instantaneous rate of change of velocity with respect time and is given by Eample : The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and eplain your choices. Section.7 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 1, 5, 7, 35, 37, 1, 3, 5 7
Section.8 - What Does f Say About f? What does f say about f? If f ()>0 on an interval, then f is If f ()<0 on an interval, then f is If f ()=0 on an interval, then f is on that interval. on that interval. on that interval. Eample 1: Given that the domain of f() is all real numbers and the graph below is of f (), determine the interval(s) on which f() is increasing and decreasing. f () a b c Eample : Let s discuss the idea of local etrema: f() a b c d e Suppose =c is in the domain of f() 1. If f () changes from to at =c, then we have that f() is and at =c there is a.. If f () changes from to at =c, then we have that f() is and at =c there is a. 3. If the sign of f () is the same on both sides of =c, then at =c. 8
Eample 3: Given that the domain of f() is all real numbers and the graph below is of f (), determine the following: y 10 8 f () 10 8 8 10 8 10 a) the interval(s) on which f() is increasing. b) the interval(s) on which f() is decreasing. c) the local etrema Eample : Discuss the difference in shapes of the graphs of f()= and g()= on [0, ) Definition: The graph of a function f is concave upward on the interval(a,b) if The graph of a function f is concave downward on the interval(a,b) if Definition: An inflection point of f occurs where where. Graphically, its 9
Eample 5: Let s look at the four basic shapes of a graph: Eample : At each point below on the graph of f() determine whether f, f, and f is positive, negative, or zero. A B D f() C Eample 7: Use the given information to sketch the graph of f. Domain: (, 1) ( 1,1) (1, ) Asymptotes: = 1, =1, y=0 f()=1, f(0)=0, f()=1 f ()>0on (, 1) (0,1) f ()<0on ( 1,0) (1, ) f ()>0 on(, 1) ( 1,1) (1, ) Section.8 Highly Suggested Homework Problems: 1, 3, 9, 15, 19, 1 10