Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative function) Homework: ü review lecture notes thoroughly ü work on exercises from the textbook in sections 2.6, 2.7, and 2.8 ü work on Assignment 5 and Assignment 6
Limits at Infinity The behaviour of functions at infinity is also known as end behaviour. What happens to the y-values of a function f(x) as the x-values increase or decrease without bounds? lim f (x) =? x lim f (x) =? x
Limits at Infinity Possibility: y-values also approach infinity or negative infinity Examples: f (x) = e x f (x) = 0.01x 3 20 15 25 10-25 0 25 50 75. 5-5 0 5 10 15 20 25 30 lime x = (limit D.N.E) x -25 lim 0.01x 3 = (limit D.N.E) x
Limits at Infinity Possibility: y-values approach a unique real number L Examples: f (x) = 3x +1 x 2 f (x) = sin x x 16 1 8 0.5-10 -5 0 5 10 15 20 25. -25 0 25 50 75 100 125 150-8 -16 lim x 3x +1 x 2 = 3-0.5 lim sin x x x = 0
Limits at Infinity Possibility: y-values oscillate and do not approach a single value Example: 1.5 1 0.5 f (x) = sin x 30-25 -20-15 -10-5 0 5 10 15 20 25 30. -0.5-1 -1.5 lim sin x D.N.E. x
Limits at Infinity Definition: lim x f (x) = L the limit of f(x), as x approaches, equals L means that the values of f(x) (y-values) can be made as close as we d like to L by taking x sufficiently large. Definition: lim f (x) = L x the limit of f(x), as x approaches, equals L means that the values of f(x) (y-values) can be made as close as we d like to L by taking x sufficiently small.
Calculating Limits at Infinity * The Limit Laws listed in 2.3 (except 9 and 10) are still valid if x a is replaced by x Theorem: If r>0 is a rational number, then 1 lim x x = 0. r If r>0 is a rational number such that 1 defined for all x, then lim x x = 0. r x r is
Calculating Limits at Infinity Examples: Find the limit or show that it does not exist. (a) 2x 2 +1 lim x 4x x 2 (b) 3x 7 lim x 2 x +1 (c) lim x x 1 x 2 + 3x + 4 (d) lim ( x + x 2 + 2x ) x
Horizontal Asymptotes Definition: The line y=l is called a horizontal asymptote of the curve y=f(x) if either lim f (x) = L x or lim x f (x) = L Example: Basic functions we know that have HAs:
Asymptotes Example: Determine the vertical and horizontal asymptotes of the rational function f (x) = 3 x x + 2.
End Behaviour Example: Determine the end behaviour of f (x) = 4x 2 x 5..
Rates of Change The rate of change of a function tells us how the dependent variable changes when there is a change in the independent variable. Geometrically, the rate of change of a function corresponds to the slope of its graph.
Slope of a Linear Function Slope = change in output change in input m = y 2 y 1 x 2 x 1 y 2 y 1 y 2 y P 2 f m = f (x 2) f (x 1 ) x 2 x 1 y 1 P 1 x 2 x 1 x 1 x 2 x
Slope of a Curve How do we calculate the slope of a curve? y y = f(x) x
Secant Lines and Tangent Lines A secant line is a line that intersects two points on a curve. A tangent line is a line that just touches a curve at a point and most closely resembles the curve at that point. y P tangent Q secant y = f(x) x
Average Rate of Change = Slope of Secant Line The average rate of change in f from x=a to x=a+h corresponds to the slope of the secant line PQ. f(a+h) f(a) y P Q secant y = f(x) a a+h x
Estimating the Slope of the Tangent Steps: 1. Approximate the tangent at P using secants intersecting P and a nearby point Q. y 2. Obtain a better approximation to the tangent at P by moving Q closer to P, but Q P. P Q 3 Q 2 Q 1 y = f(t) 3. Define the slope of the tangent at P to be the limit of the slopes of secants PQ as Q approaches P (if the limit exists): t m P = lim Q P m PQ
Estimate the slope of the tangent to f (x) = ln x Approach 1 from the left: at the base point P(1,0). Q x y = f(x) m PQ Q 1 0.5 Q 2 0.9 Q 3 0.99 Note: This is an overestimate. Approach 1 from the right: 2 1 0 1 2 3 4 5 6-1 Q x y = f(x) m PQ Q 4 1.5 Q 5 1.1 Q 6 1.01 Note: This is an underestimate. -2-3 Guess : m p =?
Instantaneous Rate of Change = Slope of Tangent Line The instantaneous rate of change of f(x) at x=a corresponds to the slope of the tangent line at x=a. f(a+h) f(a) y P tangent Q y = f(x) Note: The slope of the curve y=f(x) at P is the slope of its tangent line at P. a a+h x
Instantaneous Rate of Change = Slope of Tangent Line This special limit is called the derivative of f at a and is denoted by f (a) (read f prime of a ). Alternative formula: f(x) f(a) y P tangent Q y = f(x) a x x
Instantaneous Rate of Change = Slope of Tangent Line Example: Determine an equation of the tangent line to the curve f (x) = x 2 + 2x at P(1,3). 4 3 2 1-2 -1 0 1 2-1
Instantaneous Rate of Change = Slope of Tangent Line Example: #8. Find an equation of the tangent line to the 2x +1 curve f (x) = at the point (1,1). x + 2 10 5 10-5 0 5 10-5 -10
Velocity Suppose an object moves along a straight line according to an equation of motion s(t), where s is the displacement of the object from the origin at time t. y average velocity over the time interval t=a to t=a+h: = s(a + h) s(a) h instantaneous velocity at time t=a: s(a+h) s(a) y = s(t) v(a) = s'(a) = lim h 0 s(a + h) s(a) h. a a+h t
Velocity Example: Suppose the height of a ball thrown straight up in the air is given by the function h(t) = 40t 4.9t 2 where h is in metres and t is in seconds. (a) Find the average velocity for the time period beginning when t=2 and lasting (i) 1 second (ii) 0.1 seconds. (b) What is the velocity of the ball when t=2?
The Derivative as a Number Summary: The derivative of f at x=a, denoted by f (a), is a number given by f '(a) = lim h 0 f (a + h) f (a) h provided that this limit exists.
The Derivative as a Number Summary: * Geometrically, the number f (a) represents the slope of the tangent to the graph of f(x) at the point (a, f(a)). * The number f (a) also represents the instantanous rate of change of the function f(x) with respect to x at the exact moment when x=a.
The Derivative as a Function Definition: Given a function f(x), the derivative of f is also a function denoted by f (x) and defined as follows: df dx = f '(x) = lim h 0 f (x + h) f (x) h The domain of this function is the set of all x- values for which the limit exists. domain( f ') domain( f )
The Derivative as a Function Interpretations of f : 1. The function f (x) tells us the instantaneous rate of change of f(x) with respect to x at any value x in the domain of f (x). 2. The function f (x) tells us the slope of the tangent to the graph of f(x) at any point (x, f(x)), provided x is in the domain of f (x).
The Derivative as a Function
The Derivative as a Function Find the derivative of each function from first principles (i.e. using the limit definition). (a) y = 5x 1 (b) f (x) = x 2 + 2x (c) g(x) = 1 x 2 (d) h(x) = x + 3
The Derivative as a Function Example: (a) y = 5x 1 y x y x
The Derivative as a Function Example: (b) f (x) = x 2 + 2x y x y x
The Derivative as a Function Example: (c) g(x) = 1 x 2 4 3 2 1-4 -3-2 -1 0 1 2 3 4 2 1-4 -3-2 -1 0 1 2 3 4-1 -2
The Derivative as a Function Example: (d) h(x) = x + 3 4 3 2 1-3 -2-1 0 1 2 3 4 5 6 4 3 2 1-3 -2-1 0 1 2 3 4 5 6
Using the Derivative Example: Consider the function y = 1 x. Find equations of the tangent lines at the points (1,1) and (4, ½). 4. 3 2 1 0 1 2 3 4 5 6 7 8 9
Graphs y Example: Given the graph of f, sketch the graph of f. 2 6 f x y x
Differentiable Functions A function f(x) is said to be differentiable at x=a if we are able to calculate the derivative of the function at that point, i.e., f(x) is differentiable at x=a if exists. f '(a) = lim h 0 f (a + h) f (a) h
Differentiable Functions Geometrically, a function is differentiable at a point if its graph has a unique tangent line with a welldefined slope at that point. 3 Ways a Function Can Fail to be Differentiable: y y y x x x
Relationship Between Differentiability and Continuity Theorem: If f is differentiable at a, then f is continuous at a. If f is continuous at a, then f may or may not be differentiable at a. If f is not continuous at a, then f is not differentiable at a.