Chapter 2 Differentiation
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1 Chapter 2 Differentiation
2 SECTION 2.1 The Derivative and the Tangent Line Problem Calculus: Chapter 2 Section 2.1
3 Finding the Slope of a Secant Line m y x m sec f ( c x) f ( c) ( c x) c m sec f ( c x) f ( c) x Calculus: Chapter 2 Section 2.1
4 Definition of Tangent Line with Slope m If f is defined on an open interval containing c, and if the limit lim y f ( c x) f ( c) lim x x x0 x0 exists, then the line passing through (c, f(c)) with slope m is the tangent line to the graph of f at the point (c, f(c)). m tan Calculus: Chapter 2 Section 2.1
5 Find the equation of the tangent line at the point (1, 1) for the function: f( x) 1 x Calculus: Chapter 2 Section 2.1
6 Vertical Tangent Lines Vertical tangent lines occur at x = c when: lim x 0 f ( c x) f ( c) x lim x 0 f ( c x) f ( c) x Calculus: Chapter 2 Section 2.1
7 Calculus: Chapter 2 Section 2.1
8 Definition of the Derivative of a Function The derivative of f at x is given by f ( x x) f ( x) f '( x) lim x 0 x Provided the limit exists. For all x for which this limit exists, f is a function of x. Calculus: Chapter 2 Section 2.1
9 Definition of the Derivative of a Function The derivative of f at x is given by f ( x h) f ( x) f '( x) lim h0 h Provided the limit exists. For all x for which this limit exists, f is a function of x. Calculus: Chapter 2 Section 2.1
10 Derivative Lingo f '( x) Pronounced f prime of x Calculus: Chapter 2 Section 2.1
11 More Derivative Lingo dy dx d dx f ( x ) y ' Calculus: Chapter 2 Section 2.1
12 Finding the Derivative using Limit Process The key to finding the derivative of a function at a point or the derivative of an entire function is to eliminate the x from the denominator. Who s your new/old best friend? S.U.A. Calculus: Chapter 2 Section 2.1
13 Alternative Form of the Difference Quotient The second/alternative form of the difference quotient is: f ( x) f ( c) f '( c) lim xc x c Notice that this formula is used to evaluate the derivative at x = c. THIS FORMULA IS USED TO DETERMINE IF A FUNCTION IS DIFFERENTIABLE AT A POINT!!!! Calculus: Chapter 2 Section 2.1
14 MAJOR DIFFERENTIABILITY JUSTIFICATION: In order to prove that a function is differentiable at x = c, you must show the following: lim f ( x) f ( c) f ( x) f ( c) lim x c x c xc x c In other words, the derivative from the left side MUST EQUAL the derivative from the right side. Calculus: Chapter 2 Section 2.1
15 Derivatives Do Not Exist at the Following: Corner points Cusps Gaps, Jumps, Asymptotes, Vertical Tangents Calculus: Chapter 2 Section 2.1
16 Section 2.1: The Derivative and the Tangent Line Problem Differentiability Implies Continuity: If f is differentiable at x = c, then f is continuous at x = c. The converse of this statement is not always true (be careful). If f is not continuous at x = c, then f is not differentiable at x = c. Calculus: Chapter 2 Section 2.1
17 Find the x-values for which the slope of f ( x) 3 x is 1. Use your Calculator. Calculus: Chapter 2 Section 2.1
18 Calculus: Chapter 2 Section 2.2 SECTION 2.2 Basic Differentiation Rules
19 THE CONSTANT RULE d dx c 0 Calculus: Chapter 2 Section 2.2
20 THE POWER RULE d x n n x dx n1 Calculus: Chapter 2 Section 2.2
21 THE CONSTANT MULTIPLE RULE d c f ( x ) c f '( x ) dx Calculus: Chapter 2 Section 2.2
22 THE SUM & DIFFERENCE RULES d f ( x ) g ( x ) f '( x ) g '( x ) dx Calculus: Chapter 2 Section 2.2
23 SINE & COSINE FUNCTION d dx d dx sin( x) cos( x) cos( x) sin( x) Calculus: Chapter 2 Section 2.2
24 b( x) 3x 2 x Calculus: Chapter 2 Section 2.2
25 Find the equation of the line tangent to f ( x) 3x cos x when x = 0. Calculus: Chapter 2 Section 2.2
26 Calculus Synonyms The following expressions are all the same: Instantaneous Rate of Change Slope of a Tangent Line Derivative DO NOT CONFUSE AVERAGE RATE OF CHANGE WITH INSTANTANEOUS RATE OF CHANGE. Calculus: Chapter 2 Section 2.2
27 To Find the Average Rate of Change If you want to find the average rate of change, you simply need to find the slope of the line that connects the endpoints of the interval on which you are computing the average. You need two points of data to do this If you want to find the instantaneous rate of change, you find the derivative You only need ONE point of data to do this! Calculus: Chapter 2 Section 2.2
28 Position, Velocity, Acceleration Position, Velocity, and Acceleration are related in the following manner: Position: st () Velocity: s'( t) v( t) Acceleration: v'( t) a( t) Calculus: Chapter 2 Section 2.2
29 Position Facts Generally s(t), x(t), h(t), etc. Units = some measure of length only Displacement = change in position For example, s(b) s(a) Calculus: Chapter 2 Section 2.2
30 Velocity Facts Generally v(t) Units = distance / time (i.e., mph, m/s, cm/hr) An object is moving right/up when its velocity is greater than 0 (v(t) > 0) An object is moving left/down when its velocity is less than 0 (v(t) < 0) If v(t) = 0, the object may change direction (or may not) SPEED = ABSOLUTE VALUE OF VELOCITY Calculus: Chapter 2 Section 2.2
31 Acceleration Facts Generally a(t) Units = (distance / time) / time (i.e. m/s 2 ) Calculus: Chapter 2 Section 2.2
32 Speeding Up and Slowing Down An object is SPEEDING UP when the following occur: Algebraic: If the velocity and the acceleration agree in sign Graphical: If the curve is moving AWAY from the x-axis An object is SLOWING DOWN when the following occur: Algebraic: Velocity and acceleration disagree in sign Graphically: The velocity curve is moving towards the x- axis Calculus: Chapter 2 Section 2.2
33 A particle moves along a line so that its position at any time t > 0 is given by the function where s is given in meters and t is measured in seconds. (a)find the displacement of the particle during the first 2 seconds. (b) Find the average velocity of the particle during the first 4 seconds (c)find v(t) and a(t) Calculus: Chapter 2 Section s( t) t 4t 3
34 A particle moves along a line so that its position at any time t > 0 is given by the function where s is given in meters and t is measured in seconds. (d) Find the velocity of the particle at t = 4 (e) Find the acceleration of the particle at t = 4 (f) At what time(s) does the particle change direction? (g) Is the particle speeding up or slowing down at t = 3? Calculus: Chapter 2 Section s( t) t 4t 3
35 Calculus: Chapter 2 Section 2.2
36 Calculus: Chapter 2 Section 2.2
37 SECTION 2.3 Product, Quotient Rule, Higher-Order Derivatives Calculus: Chapter 2 Section 2.3
38 THE PRODUCT RULE d f ( x ) g ( x ) f ( x ) g '( x ) f '( x ) g ( x ) dx or the easier version: d u v u v v u dx ' ' Calculus: Chapter 2 Section 2.3
39 f ( x) (3x 2 x )(5 4 x) 2 Calculus: Chapter 2 Section 2.3
40 f ( x) 2xsin x 2sin x Calculus: Chapter 2 Section 2.3
41 f( x) 3x x 1 Calculus: Chapter 2 Section 2.3
42 THE QUOTIENT RULE d u vu ' uv ' dx v v 2 Calculus: Chapter 2 Section 2.3
43 THE QUOTIENT RULE Lo-d-Hi minus Hi-d-lo lo 2 Calculus: Chapter 2 Section 2.3
44 More Trig Functions: d dx tan x sec 2 x d dx sec x sec xtan x d dx cot x csc 2 x d dx csc x csc xcot x Calculus: Chapter 2 Section 2.3
45 Higher Order Derivatives are derivatives of the derivative, etc. 1 st Derivative: 2 nd Derivative: 3 rd Derivative: f f f '( x) or dy dx 2 ''( x) or d y dx 2 3 '''( x) or d y dx 3 Calculus: Chapter 2 Section 2.3
46 Calculus: Chapter 2 Section 2.4 SECTION 2.4 The Chain Rule
47 Who s the Mother Function? y sin(3 x) y 1 2x 3 3 y 3x8 4 ycsc 2 2x y 3 x 3 4x Calculus: Chapter 2 Section 2.4
48 THE CHAIN RULE If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x, and df df du dx du dx Calculus: Chapter 2 Section 2.4
49 Another Way d f ( g ( x )) f ' g ( x ) g '( x ) dx Calculus: Chapter 2 Section 2.4
50 2 3 f ( x) 3x 2x Calculus: Chapter 2 Section 2.4
51 g x x 2 2 ( ) 3 1 Calculus: Chapter 2 Section 2.4
52 ht () 7 2t 3 2 Calculus: Chapter 2 Section 2.4
53 Street Version Identify the MOTHER FUNCTION and the COMPOSED FUNCTION Differentiate the MOTHER FUNCTION Differentiate the COMPOSED FUNCTION Multiply the resultant derivatives Calculus: Chapter 2 Section 2.4
54 f( x) x 3 2 x 4 Calculus: Chapter 2 Section 2.4
55 f ( x) x 1x 2 2 Calculus: Chapter 2 Section 2.4
56 gx ( ) 3x 1 2 x 3 2 Calculus: Chapter 2 Section 2.4
57 y 2 sin( x 1) Calculus: Chapter 2 Section 2.4
58 y 2 cos x Calculus: Chapter 2 Section 2.4
59 y sin(2 x) Calculus: Chapter 2 Section 2.4
60 y 2 tan (3 x) Calculus: Chapter 2 Section 2.4
61 y 53x y ' x y 3tan(4 x) y ' 12sec 2 (4 x) Calculus: Chapter 2 Section 2.4
62 1 y x 16 x y ' x 16 x 2 x x 2 Calculus: Chapter 2 Section 2.4
63 Find y (0) for y x y "(0) 128 Calculus: Chapter 2 Section 2.4
64 Calculus: Chapter 2 Section 2.4
65 Calculus: Chapter 2 Section 2.5 SECTION 2.5 Implicit Differentiation
66 Implicit/Explicit Functions Explicit functions are typically written in a y= format (the function is written in terms of x). Implicit functions/relations are typically written in terms of both x s and y s. Calculus: Chapter 2 Section 2.5
67 Implicit Differentiation Implicit differentiation is essentially a glorified CHAIN RULE. For example: d dx y 3 3 y 2 dy dx Calculus: Chapter 2 Section 2.5
68 Rules for Implicit Differentiation 1. Differentiate both sides of the equation with respect to x 2. Collect all terms involving dy/dx on the left side of the equation, and move the other terms to the right side. 3. Factor dy/dx out of the left side 4. Solve for dy/dx Calculus: Chapter 2 Section 2.5
69 Find dy/dx given that y y 5y x 4 Calculus: Chapter 2 Section 2.5
70 Determine the slope of the tangent line to the graph of x 2 4y 2 4 at the point 2,. 1 2 Calculus: Chapter 2 Section 2.5
71 Determine the slope of the graph of x y 100xy 3,1 at the point. Calculus: Chapter 2 Section 2.5
72 Find dy/dx for the equation sin y x Calculus: Chapter 2 Section 2.5
73 Find 2 d y dx 2 for the equation x 2 2 y 25 Calculus: Chapter 2 Section 2.5
74 Calculus: Chapter 2 Section 2.5
75 Calculus: Chapter 2 Section 2.5
76 Calculus: Chapter 2 Section 2.6 SECTION 2.6 Related Rates
77 Related Rates Equations Related Rates problems are essentially implicit differentiation word problems. All functions/equations are differentiated with respect to time. Calculus: Chapter 2 Section 2.6
78 WHAT IS CHANGING? WHAT IS CONSTANT? Calculus: Chapter 2 Section 2.6
79 Calculus: Chapter 2 Section 2.6
80 Related Rates Guidelines 1. G: Identify all given quantities. 2. F: Identify all quantities to find. 3. E: Write an equation involving the variables whose rates of change are either given or are to be determined 4. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to t. 5. Solve for the desired rates of change Calculus: Chapter 2 Section 2.6
81 Suppose x and y are both differentiable functions of t and are related by 2 the equation y x 3. Find dy/dt when x = 1, given that dx/dt = 2 when x = 1. Calculus: Chapter 2 Section 2.6
82 A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius, r, of the outer ripple is increasing at a rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area of the disturbed water changing? Calculus: Chapter 2 Section 2.6
83 Air is being pumped into a spherical balloon at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet. V 4 r 3 3 Calculus: Chapter 2 Section 2.6
84 A teacher has decided to purchase a water cooler for a classroom. In order to properly hydrate the students, he has purchased conical cups from which the students will sip. Each cup has a height of 4 and radius of 1. The water cooler dispenses water into the cups at a rate of 0.5 cubic inches per minute. Determine the rate at which the water level is rising when the height of the water in the cup is V r h 3 Calculus: Chapter 2 Section 2.6
85 An airplane is flying on a flight path that will take it directly over a radar tracking station, as shown in the diagram below. If s is decreasing at a rate of 400 mph when s = 10 miles, what is the speed of the plane? s 6 mi. Calculus: Chapter 2 Section 2.6
86 A television camera at ground level is filming the lift-off of a space shuttle that is rising vertically according to the position equation s=50t 2, where s is measured in feet and t is measured in seconds. The camera is 2000 feet from the launch pad. Find the rate of change in the angle of elevation of the camera shown below at 10 seconds after lift-off feet Calculus: Chapter 2 Section 2.6
87 Find the rate of change in the angle of elevation of the camera shown below at 10 seconds after lift-off feet Calculus: Chapter 2 Section 2.6
88 HW #26 A trough is 12 feet long and 3 feet across the top. Its ends are isosceles triangles with altitudes of 3 feet. 12 feet 3 feet 3 feet (a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when h is 1 foot deep? (b) If the water is rising at a rate of 3/8 inches per minute when h = 2 feet, determine the rate at which water is being pumped into the trough. Calculus: Chapter 2 Section 2.6
89 12 feet 3 feet 3 feet Calculus: Chapter 2 Section 2.6
90 HW #27 A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. y x (a) How fast is the top of the ladder moving down the wall when its base is 7 feet? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall. Calculus: Chapter 2 Section 2.6
91 Calculus: Chapter 2 Section
92 7 12 A 1 x 2 1 A 2 y x y 1 2 x dy dy y dx dt Calculus: Chapter 2 Section 2.6
93 Calculus: Chapter 2 Section 2.6
94 HW # A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat. (a) The winch pulls the rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock? Calculus: Chapter 2 Section 2.6
95 Calculus: Chapter 2 Section 2.6
96 Calculus: Chapter 2 Section 2.6
97 HW # A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) At what rate is the tip of the shadow moving? (b) At what rate is the length of his shadow changing? Calculus: Chapter 2 Section 2.6
98 Calculus: Chapter 2 Section 2.6
99 HW # A security camera is centered 50 feet above a 100-foot hallway. It is easiest to design the camera with a constant angular rate of rotation, but this results in a variable rate at which the images of the surveillance area are recorded. So, it is desirable to design a system with a variable rate of rotation and a constant rate of movement of the scanning beam along the hallway. Find a model for the variable rate of rotation if dx / dt 2 feet per second Calculus: Chapter 2 Section 2.6
100 Calculus: Chapter 2 Section 2.6
MTH 125 3.7 Related Rates
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