Chapter 3 Derivatives

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1 Chapter Derivatives Section 1 Derivative of a Function What you ll learn about The meaning of differentiable Different ways of denoting the derivative of a function Graphing y = f (x) given the graph of y = f (x) Graphing y = f (x) given the graph of y = f (x) One-sided derivatives Graphing the derivative from data and why The derivative gives the value of the slope of the tangent line to a curve at a point Copyright 015, 01, and 009 Pearson Education, Inc 1 Copyright 016, 01, and 010 Pearson Education, Inc Definition of Derivative Differentiable Function The derivative of the function f with respect to the variable x is the function f ( x) whose value at x is f ( x + h) - f ( x) f ( x) = lim h 0 h provided the limit exists The domain of f, the set of points in the domain of f for which the limit exists, may be smaller than the domain of f If f ( x) exists, we say that f has a derivative (is differentiable) at x A function that is differentiable at every point in its domain is a differentiable function Copyright 016, 01, and 010 Pearson Education, Inc Copyright 016, 01, and 010 Pearson Education, Inc 4 Example Definition of Derivative Derivative at a Point (alternate) The derivative of the function f at the point where x = a is the limit f ( x) - f ( a) f ( a) = lim x a x - a provided the limit exists Copyright 016, 01, and 010 Pearson Education, Inc 5 Copyright 016, 01, and 010 Pearson Education, Inc 6 1

2 Derivative of a Function at Point Notation There are many ways to denote the derivative of a function y = f ( x) Besides f '( x), the most common notations are: y y prime Nice and brief, but does not name the independent variable dy dy or the derivative Names both variables and of y with respect to x uses d for derivative df df or the derivative of f with respect to x Emphasizes the function s name d f ( x ) d of f at x or the Emphasis that differentiation derivative of f at x is an operation performed on f Copyright 016, 01, and 010 Pearson Education, Inc 7 Copyright 016, 01, and 010 Pearson Education, Inc 8 Relationships between the Graphs of f and f Example 4 Because we can think of the derivative at a point in graphical terms as slope, we can get a good idea of what the graph of the function f looks like by estimating the slopes at various points along the graph of f We estimate the slope of the graph of f in y-units per x-unit at frequent intervals We then plot the estimates in a coordinate plane with the horizontal axis in x-units and the vertical axis in slope units Copyright 016, 01, and 010 Pearson Education, Inc 9 Copyright 016, 01, and 010 Pearson Education, Inc 10 Graphing the Derivative from Data Discrete points plotted from sets of data do not yield a continuous curve, but we have seen that the shape and pattern of the graphed points (called a scatter plot) can be meaningful nonetheless It is often possible to fit a curve to the points using regression techniques If the fit is good, we could use the curve to get a graph of the derivative visually However, it is also possible to get a scatter plot of the derivative numerically, directly from the data, by computing the slopes between successive points One-sided Derivatives A function y = f ( x) is differentiable on a closed interval a, b if it has a derivative at every interior point on the interval, and if the limits f ( a + h) - f ( a) lim the right - hand derivative at a + h 0 h f ( b + h) - f ( b) lim the left - hand derivative at b - h 0 h exist at the endpoints In the right-hand derivative, h is positive and a + h approaches a from the right In the left-hand derivative, h is negative and b + h approaches b from the left Copyright 016, 01, and 010 Pearson Education, Inc 11 Copyright 016, 01, and 010 Pearson Education, Inc 1

3 One-sided Derivatives Example One-sided Derivatives Right-hand and left-hand derivatives may be defined at any point of a function s domain Show that the following function has left-hand and right-hand derivatives at x = 0, but no derivative there The usual relationship between one-sided and two-sided limits holds for derivatives Theorem, Section 1, allows us to conclude that a function has a (two-sided) derivative at a point if and only if the function s right-hand and left-hand derivatives are defined and equal at that point AP Calculus -x, x 0 y = x, x 0 Chapter Derivatives Copyright 016, 01, and 010 Pearson Education, Inc 1 Copyright 016, 01, and 010 Pearson Education, Inc 14 Chapter Derivatives Section Differentiability What you ll learn about Why f (a) might fail to exist at x = a Differentiability implies local linearity Numerical derivatives on a calculator Differentiability implies continuity Intermediate Value Theorem for derivatives and why Graphs of differentiable functions can be approximated by their tangent lines at points where the derivative exists Copyright 015, 01, and 009 Pearson Education, Inc 15 Copyright 016, 01, and 010 Pearson Education, Inc 16 How f (a) Might Fail to Exist A function will not have a derivative at a point P ( a, f ( a) ) f ( x) - f ( a) where the slopes of the secant lines, x - a fail to approach a limit as x approaches a The next figures illustrate four different instances where this occurs For example, a function whose graph is otherwise smooth will fail to have a derivative at a point where the graph has: How f (a) Might Fail to Exist 1 a corner, where the one-sided derivatives differ; f ( x) = x Copyright 016, 01, and 010 Pearson Education, Inc 17 Copyright 016, 01, and 010 Pearson Education, Inc 18

4 How f (a) Might Fail to Exist How f (a) Might Fail to Exist a cusp, where the slopes of the secant lines approach from one side and approach - from the other (an extreme case of a corner); f ( x) = x A vertical tangent, where the slopes of the secant lines approach either or - from both sides; f ( x) = x Copyright 016, 01, and 010 Pearson Education, Inc 19 Copyright 016, 01, and 010 Pearson Education, Inc 0 How f (a) Might Fail to Exist Example How f (a) Might Fail to Exist 4 a discontinuity (which will cause one or both of the one-sided derivatives to be nonexistent) -1, x 0 U ( x) = 1, x 0 Show that the function is not differentiable at x = 0 x ( ), x 0 f x = 4 x, x 0 Copyright 016, 01, and 010 Pearson Education, Inc 1 Copyright 016, 01, and 010 Pearson Education, Inc Differentiability Implies Local Linearity Differentiability Implies Local Linearity A good way to think of differentiable functions is that they are locally linear; that is, a function that is differentiable at a closely resembles its own tangent line very close to a In the jargon of graphing calculators, differentiable curves will straighten out when we zoom in on them at a point of differentiability Copyright 016, 01, and 010 Pearson Education, Inc Copyright 016, 01, and 010 Pearson Education, Inc 4 4

5 Numerical Derivatives on a Calculator Many graphing utilities can approximate derivatives numerically with good accuracy at most points of their domains For small values of h, the difference quotient f ( a + h) - f ( a) h is often a good numerical approximation of f ( a) However, the same value of h will usually yield a better approximation if we use the symmetric difference quotient f ( a + h) - f ( a - h) h which is what our graphing calculator uses to calculate NDER f ( a), the numerical derivative of f at a point a The numerical derivative of f as a function is denoted by NDER f ( x) The numerical derivatives we compute in this book will use h = 0001 Numerical Derivatives on a Calculator The numerical derivative of f at a, which we will denote NDER ( f ( x),a), is the number ( ) - f ( a ) f a The numerical derivative of f, which we will denote NDER ( f ( x), x), is the function ( ) - f ( x ) f x Copyright 016, 01, and 010 Pearson Education, Inc 5 Copyright 016, 01, and 010 Pearson Education, Inc 6 Example Derivatives on a Calculator Find the numerical derivative of the function f ( x) = x + at the point x = Use a calculator with h = 0001 Derivatives on a Calculator Because of the method used internally by the calculator, you will sometimes get a derivative value at a nondifferentiable point This is a case of where you must be smarter than the calculator Copyright 016, 01, and 010 Pearson Education, Inc 7 Copyright 016, 01, and 010 Pearson Education, Inc 8 Differentiability Implies Continuity Intermediate Value Theorem for Derivatives If f has a derivative at x = a, then f is continuous at x = a Not every function can be a derivative The converse of Theorem 1 is false A continuous functions might have a corner, a cusp or a vertical tangent line, and hence not be differentiable at a given point If a and b are any two points in an interval on which f is differentiable, then f takes on every value between f ( a) and f ( b) Copyright 016, 01, and 010 Pearson Education, Inc 9 Copyright 016, 01, and 010 Pearson Education, Inc 0 5

6 Chapter Derivatives Section Rules for Differentiation What you ll learn about Power functions Sum and difference rules Product and quotient rules Negative integer powers of x Second and higher order derivatives and why These rules help us find derivatives of functions analytically in a more efficient way Copyright 015, 01, and 009 Pearson Education, Inc 1 Copyright 016, 01, and 010 Pearson Education, Inc Rule 1 Derivative of a Constant Function Rule Power Rule for Positive Integer Powers of x If f is the function with the constant value c, then df d = ( c ) = 0 This means that the derivative of every constant function is the zero function If n is a positive integer, then d x nx The Power Rule says: n n-1 ( ) = n To differentiate x, multiply by n and subtract 1 from the exponent Find the derivative of Find the derivative of Copyright 016, 01, and 010 Pearson Education, Inc Copyright 016, 01, and 010 Pearson Education, Inc 4 Rule The Constant Multiple Rule Rule 4 The Sum and Difference Rule If u is a differentiable function of x and c is a constant, then d du ( cu ) = c This says that if a differentiable function is multiplied by a constant, then its derivative is multiplied by the same constant If u and v are differentiable functions of x, then their sum and differences are differentiable at every point where u and v are differentiable At such points, d du dv ( u v) = Find the derivative of Find the derivative of Copyright 016, 01, and 010 Pearson Education, Inc 5 Copyright 016, 01, and 010 Pearson Education, Inc 6 6

7 Example Positive Integer Powers, Multiples, Sums, and Differences Differentiate the polynomial Example Positive Integer Powers, Multiples, Sums, and Differences 4 Does the curve y = x - 8x + have any horizontal tangents? If so, where do they occur? Verify you result by graphing the function Copyright 016, 01, and 010 Pearson Education, Inc 7 Copyright 016, 01, and 010 Pearson Education, Inc 8 Rule 5 The Product Rule The product of two differentiable functions u and v is differentiable, and d dv du ( uv) = u + v The derivative of a product is actually the sum of two products Example Using the Product Rule ( ) ( ) = ( - )( + ) Find f x if f x x 4 x Copyright 016, 01, and 010 Pearson Education, Inc 9 Copyright 016, 01, and 010 Pearson Education, Inc 40 Rule 6 The Quotient Rule Example Using the Quotient Rule u At a point where v 0, the quotient y = of two differentiable v functions is differentiable, and du dv v - u d u = v v Since order is important in subtraction, be sure to set up the numerator of the Quotient rule correctly Find x - 4 f ( x) if f ( x) = x + Copyright 016, 01, and 010 Pearson Education, Inc 41 Copyright 016, 01, and 010 Pearson Education, Inc 4 7

8 Rule 7 Power Rule for Negative Integer Powers of x If n is a negative integer and x 0, then d 1 ( x n ) nx n- = This is basically the same as Rule except now n is negative Example Negative Integer Powers of x 1 Find an equation for the line tangent to the curve y = at the point ( 1,1 ) x Copyright 016, 01, and 010 Pearson Education, Inc 4 Copyright 016, 01, and 010 Pearson Education, Inc 44 Second and Higher Order Derivatives Second and Higher Order Derivatives dy The derivative y = is called the first derivative of y with respect to x The first derivative may itself be a differentiable function of x If so, dy d dy d y its derivative, y = = =, is called the second derivative of y with respect to x If y ( y ) double prime is differentiable, its derivative, dy d y y = =, is called the third derivative of y with respect to x The multiple-prime notation begins to lose its usefulness after three primes ( n) d ( n-1) So we use y = y y super n to denote the nth derivative of y with respect to x ( n) n Do not confuse the notation y with the nth power of y, which is y Copyright 016, 01, and 010 Pearson Education, Inc 45 Copyright 016, 01, and 010 Pearson Education, Inc 46 Finding Higher Order Derivatives Chapter Derivatives Section 4 Velocity and Other Rates of Change Copyright 016, 01, and 010 Pearson Education, Inc 47 Copyright 015, 01, and 009 Pearson Education, Inc 48 8

9 What you ll learn about Instantaneous rates of change Motion on a line Acceleration as the second derivative Modeling vertical motion and particle motion The derivative as a measure of sensitivity to change Marginal cost and marginal revenue Instantaneous Rates of Change The instantaneousrate of change of f with respect to x at a is the derivative ( ) - f ( a) f a + h f ( a)= lim h 0 h provided the limit exists When we say rate of change, we mean instantaneous rate of change and why Derivatives give the rates at which things change in the world Copyright 016, 01, and 010 Pearson Education, Inc 49 Copyright 016, 01, and 010 Pearson Education, Inc 50 Example Instantaneous Rates of Change Motion Along a Line If the area of a square as a function of the radius is A= r, ( a) Find the rate of change of the area A with respect to the radius r ( b) Evaluate the rate of change of A when r = 4 Suppose that an object is moving along a coordinate line so that we know its position s on that line as a function of time t: s = f ( t) The displacement of the object over the time interval from t to t + t is s = f ( t + t ) - f ( t) The average velocity of the object over that time interval is displacement s f ( t + t ) - f ( t) vav = = = travel time t t Copyright 016, 01, and 010 Pearson Education, Inc 51 Copyright 016, 01, and 010 Pearson Education, Inc 5 Instantaneous Velocity Example The ( instantaneous ) velocity is the derivative of the position function s = f ( t) with respect to time At time t the velocity is ds f ( t + t ) - f ( t ) v ( t ) = = lim dt t 0 t Copyright 016, 01, and 010 Pearson Education, Inc 5 Copyright 016, 01, and 010 Pearson Education, Inc 54 9

10 Speed Speed is the absolute value of velocity ds Speed = v( t ) = dt Example The graph shows the velocity of a particle moving on a coordinate line 1 When does the particle move forward? backward? Speed up? Slow down? When is the particle s acceleration positive? Negative? Zero? When does the particle move at its greatest speed? 4 When does the particle stand still for more than an instant? Copyright 016, 01, and 010 Pearson Education, Inc 55 Copyright 016, 01, and 010 Pearson Education, Inc 56 Acceleration Free-fall Constants (Earth) Acceleration is the derivative of velocity with respect to time ds If a body s velocity at time t is v( t) = then the body s dt dv d s acceleration at time t is a ( t ) = = dt dt English Units Metric Units g = ft sec, s= 1 ( )t =16t (s in feet) g =98 m sec, s= 1 ( 98)t = 49t (t in meters) Copyright 016, 01, and 010 Pearson Education, Inc 57 Copyright 016, 01, and 010 Pearson Education, Inc 58 Example Finding Velocity A projectile is shot upward from the surface of the earth and reaches a height of s =- 49t + 10 t meters after t seconds Find the velocity after 5 seconds Example 5 Copyright 016, 01, and 010 Pearson Education, Inc 59 Copyright 016, 01, and 010 Pearson Education, Inc 60 10

11 Sensitivity to Change Derivatives in Economics When a small change in x produces a large change in the value of a function f (x), we say that the function is relatively sensitive to changes in x The derivative f (x) is a measure of this sensitivity Economists have a specialized vocabulary for rates of change and derivatives They call them marginals In a manufacturing operation, the cost of production c(x) is a function of x, the number of units produced The marginal cost of production is the rate of change of dc cost with respect to the level of production, so it is Sometimes the marginal cost of production is loosely defined to be the extra cost of producing one more unit Copyright 016, 01, and 010 Pearson Education, Inc 61 Copyright 016, 01, and 010 Pearson Education, Inc 6 Example Derivatives in Economics Suppose that the dollar cost of producing x washing machines is c( x) = x - 01 x Find the marginal cost of producing 100 washing machines Chapter Derivatives Section 5 Derivatives of Trigonometric Functions Copyright 016, 01, and 010 Pearson Education, Inc 6 Copyright 015, 01, and 009 Pearson Education, Inc 64 What you ll learn about Derivative of the Sine and Cosine Functions Derivatives of the sine and cosine functions Modeling harmonic motion Jerk as the derivative of acceleration Derivatives of the tangent, cotangent, secant, and cosecant functions Tangent and normal lines and why The derivatives of sines and cosines play a key role in describing periodic change The derivative of the sine is the cosine d sin x = cos x The derivative of the cosine is the negative of the sine d cos x = - sin x Copyright 016, 01, and 010 Pearson Education, Inc 65 Copyright 016, 01, and 010 Pearson Education, Inc 66 11

12 Example Finding the Derivative of the Sine and Cosine Functions sin x Find the derivative of ( cos x - ) Simple Harmonic Motion The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion Copyright 016, 01, and 010 Pearson Education, Inc 67 Copyright 016, 01, and 010 Pearson Education, Inc 68 Example Simple Harmonic Motion Jerk A weight hanging from a spring bobs up and down with position function s = sin t ( s in meters, t in seconds ) What are its velocity and acceleration at time t? Jerk is the derivative of acceleration If a body s position at time t is da d s j ( t) = dt = dt Copyright 016, 01, and 010 Pearson Education, Inc 69 Copyright 016, 01, and 010 Pearson Education, Inc 70 Derivative of the Other Basic Trigonometric Functions d tan x = sec x d cot x = - csc x d sec x = sec x tan x d csc x = - csc xcot x Example Derivative of the Other Basic Trigonometric Functions Find the equation of a line tangent to y = xcos x at x= 1 Copyright 016, 01, and 010 Pearson Education, Inc 71 Copyright 016, 01, and 010 Pearson Education, Inc 7 1

13 Example 5 Copyright 016, 01, and 010 Pearson Education, Inc 7 1