Announcements Topics: Work On: Sections 3.1, 3.2, and 3.3 * Read these sections and study solved examples in your textbook! Work On: Assignments 8, 9, and part of 10 Exercises in the textbook as assigned in the coursepack

Calculus On Continuous Functions In order to start studying continuous-time dynamical systems, we need to develop the usual tools of calculus on continuous functions: Limits Continuity these are all defined Derivatives in terms of limits Integrals 3.1 THE LIMIT IS THE MOST IMPORTANT CONCEPT IN CALCULUS, AS ALL THE FUNDAMENTAL IDEAS OF CALCULUS – CONTINUITY, DERIVATIVE, INTEGRAL – ARE BASED ON IT (DEFINED IN TERMS OF IT)

Rates of Change The rate of change of a function tells us how fast 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 it’s graph. First discovery of the necessity of the idea of a limit. How do we find the slope of a straight line? Easy… it is constant… how do we define the slope of a curve? HARDER… we define two types of lines

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. The slope of a straight line is easy to define – change in output (delta y) over change in input (delta x). – write it How do we define the slope of a curve? 2 ideas: slope of a secant Tangent=best straight line approximate to the curve… has same point and slope as the curve ZOOM IN

Average Rate of Change = Slope of Secant Line The average rate of change of f(t) from t=t1 to t=t2 corresponds to the slope of the secant line PQ. Change to notation in the book… really, this is just y2-y1/x2-x1…UNITS: UNITS OF DEPENDENT/UNIT OF INDEPENDENT

Average Rate of Change = Slope of Secant Line Alternative Notation: The average rate of change of f(t) from the base point t=t0 to t=t0+Δt is Change to notation in the book… really, this is just y2-y1/x2-x1…UNITS: UNITS OF DEPENDENT/UNIT OF INDEPENDENT

Average Rate of Change = Slope of Secant Line Example #2 (modified): Find the average rate of change of the function starting from time t=1 and lasting 1, 0.1, and 0.01 units of time. . 2.1 2.01 2.001 t0 Δt t0+Δt f(t0+Δt) - f(t0) 1 2 0.1 1.1 0.01 1.01 So t=1 is called the “basepoint” and the duration is the time step or delta t…

Estimating the Slope of the Tangent Steps: 1. Approximate the tangent at P using secants intersecting P and a nearby point Q. 2. Obtain a better approximation to the tangent at P by moving Q closer to P, but Q P. 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). Write this out in words… lim Q->P…

Average Rate of Change = Slope of Secant Line Example #10 (modified): (a) Guess the limit of the slopes of the secants as the second point approaches the base point. (b) Use this to find the equation of the tangent line to h(t) at t=1. . t0 Δt t0+Δt f(t0+Δt) - f(t0) 1 0.1 1.1 2.1 0.01 1.01 2.01 0.001 1.001 2.001 Change to notation in the book… really, this is just y2-y1/x2-x1…UNITS: UNITS OF DEPENDENT PER UNIT OF INDEPENDENT

Instantaneous Rate of Change = Slope of Tangent Line The instantaneous rate of change of f(t) at t=t0 corresponds to the slope of the tangent line at t=t0. Note: The slope of the curve y=f(t) at P is the slope of its tangent line at P. Different notation – prime notation vs differential notation Also show that this is lim as change in x->0 of change in y/change in x

Instantaneous Rate of Change = Slope of Tangent Line This special limit is called the derivative of f at t0 and is denoted by f’(t0) (read “f prime of t0”). Alternative notation: Different notation – prime notation vs differential notation Also show that this is lim as change in x->0 of change in y/change in x

The Limit of a Function Notations: The value of a function at a particular value of its independent variable was defined first. We saw with the “tangent problem”, i.e., finding the slope of a tangent line to a curve when we only have one point on which to base the slope, that we need the idea of a limit. A limit is the second half of the story. So we can ask, what is the value of a function AT a point and we can also ask, what is happening to the y-values, what is the pattern/trend in the y-values of the function f(x) as x gets closer and closer to some number a.

The Limit of a Function Definition: We say that the limit of f(x) as x approaches a equals L, and write if we can make the values of f(x) as close to L as desired by taking x sufficiently close to a, but not equal to a. Talk about alternative notations… f->L as x->a …. Read arrows as “approaches” F(x) gets closer and closer to L as the numbers x get closer and closer to a (from either side), but x is not equal to a.

Limit of a Function Some examples: Ask about the value of the limit and value of the function at each key point. Last one: Since infinity is not a real number, we say that the limit DNE. Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a.

Left-Hand and Right-Hand Limits means as from the left means as from the right ** The full limit exists if and only if the left and right limits both exist (equal a real number) and are the same value. We often need these definitions when dealing with PIECEWISE functions Also, with root functions and logarithmic functions

Left-Hand and Right-Hand Limits For each function below, determine the value of the limit or state that it does not exist. BREAK EACH ONE DOWN INTO LEFT AND RIGHT LIMITS. FOR THE MIDDLE ONE, MENTION HOW WE CAN ONLY FIND THE RIGHT LIMIT BECAUSE G IS NOT DEFINED TO THE LEFT OF 0. FOR THE LAST ONE, STRESS THAT THE LIMIT EXISTS EVEN THOUGH THE FUNCTION IS NOT DEFINED AT X=3.

Evaluating Limits We can evaluate the limit of a function in 3 ways: Graphically Numerically Algebraically Explain each…. In 2.2, we will just do them graphically and numerically… in 2.3, we will learn how to evaluate them algebraically

Evaluating Limits Example: Evaluate graphically. Graph it, do little arrows along the x-axis, y-axis, and on the curve like the textbook does. CAN SKIP>

Evaluating Limits Example: Use a table of values to estimate the value of x f(x) 3.5 3.9 3.99 4 undefined 4.01 4.1 4.5 Guessing based on a partial table of values can produce WRONG answers… see examples in text. But it is a starting out point.

LIMIT LAWS [used to evaluate limits algebraically] Suppose that c is a constant and the limits exist. Then 1. 2. 3. CHANGE These can be understood from an intuitive point of view….read Example 1 in the textbook… nice illustration of these laws when the functions f and g are given by graphs.

LIMIT LAWS [used to evaluate limits algebraically] Continued… 4. 5. 6. 7. 8. Constant function, identity function. 7 and 8 can be understood on an intuitive level or by looking at the graph. 6. is obtained by applying product law (4) repeatedly with g(x)=f(x).

LIMIT LAWS [used to evaluate limits algebraically] Continued… 9. 10. 11. 9. Put f(x)=x in Law 6 and use Law 8.

Evaluating Limits Algebraically Example: Evaluate the limit algebraically and justify each step by indicating the appropriate Limit Laws. Read example 1 in the text when they use limit laws and read limits from a graph.

Direct Substitution Property From the previous slide, we have Notice that we could have simply found the value of the limit by plugging in x=1 into the function. WHAT FUNCTIONS WILL DIRECT SUBSTITUTION WORK FOR? WELL… ALL FUNCTIONS CONTINUOUS AT X=A BUT WE WILL LEARN THAT NEXT SECTION…. BTW, PROOFS OF ALL PROPERTIES ARE DONE USING THE “PRECISE DEFN OF A LIMIT” BUT WE ARE SKIPPING THESE FORMAL PROOFS WHICH ARE OFTEN DONE IN A REAL ANALYSIS CLASS.

Direct Substitution Property If f is an algebraic, exponential, logarithmic, trigonometric, or inverse trigonometric function, and is in the domain of f, then This one is different.

Equal Limits Property Consider the functions: Note; direct subs doesn’t work on the left one but it DOES work on the right one. What do they have in common? They both have the same value of the limit as x approaches 2 (value of the limit is 4). We calculate the limit of the f by simplifying it to g and then finding the limit of g. * Note: f(x)=g(x) everywhere except at x=2

Equal Limits Property Example: Calculate FACT: If when , then provided the limits exist. Note: direct substitution does not work We can compute the limit of f by replacing it with a simpler function g with the same limit. This is valid because f=g except when x=2 and in computing a limit as x approaches 2 we are not concerned about what is happening when x actually equals 2.

Strategy for Evaluating Limits

Evaluating Limits Algebraically Evaluate each limit or state that it does not exist. (a) (b) (c) (d)

Infinite Limits Example: Use a table of values to estimate the value of x f(x) 0.1 0.01 0.001 undefined -0.001 -0.01 -0.1 Size of the y-values is unbounded… as x->0 from right, y-values continue to grow without bounds… we could make y as large as we would like by choosing x sufficiently close to 0, but not =0, on the right of x=0. Also, GRAPH. NOTE: A NON-ZERO NUMBER DIVIDED BY ZERO = PLUS/MINUS INFINITY

Infinite Limits Definition: “the limit of f(x), as x approaches a, is infinity” means that the values of f(x) (y-values) increase without bound as x becomes closer and closer to a (from either side of a), but x a. Definition: “the limit of f(x), as x approaches a, is negative infinity” means that the values of f(x) (y-values) decrease without bound as x becomes closer and closer to a (from either side of a), but x a. Infinity is a concept, not a number. MEANS THAT WE CAN MAKE THE Y-VALUES AS LARGE AS WE’D LIKE BY CHOOSING THE X-VALUES SUFFICIENTLY CLOSE TO, BUT NOT =, A value is infinitely large if it exceeds the capacity of every possible measuring device. .

Infinite Limits Example: Determine the infinite limit. (a) (b) Note: Since the values of these functions do not approach a real number L, these limits do not exist. Reitterate that NONE OF THESE LIMITS EXISIT! BE DETERMINING THESE LIMITS, WE ARE JUST EXPLAING WHY THE LIMIT DOESN’T EXISIT… GIVING MORE INFORMATION AND BEING MORE SPECIFIC ABOUT THE NATURE OF IT… ALSO, IT HELPS US TO UNDERSTAND THE GRAPH (VAs)

Vertical Asymptotes Definition: The line x=a is called a vertical asymptote of the curve y=f(x) if either Example: Basic functions we know that have VAs: Mention important functions that have VAs.... So as we approach the values of x not in their domain, the values of the function increase or decrease without bounds so we have an infinite limit. Ln x, tanx, 1/x, 1/x^2

Limits at Infinity The behaviour of functions “at” infinity is also known as the end behaviour or long-term behaviour of the function. What happens to the y-values of a function f(x) as the x-values increase or decrease without bounds? END BEHAVIOUR of a function… we can evaluate graphically or algebraically. WHAT HAPPENS TO THE GRAPH REALLY FAR TO THE RIGHT OR REALLY FAR TO THE LEFT? WHY? We would sometimes like to know the long term behaviour of some function… so if we have a function that measures the concentration of toxic chemicals in the lungs as a function of time, we might want to know what will happen long term? Ie as t-> infinity I know in high school they teach to to plug in a very large value and this works sometimes but is not a reliable way to analyze functions.

Limits at Infinity Possibility: y-values also approach infinity or - infinity Examples: . Limit as x->infinity DNE… what happens to the y-values as we go really far to the left or to the right of the graph?

Limits at Infinity Possibility: y-values approach a unique real number L Examples: . Limit as x-> infinity exists! The farther out you get on either side of the x-axis, the closer the y-values get to a single number. In this case, we say that the values of the function become indistinguishable from this number L as the x-values increase without bounds.

Limits at Infinity Possibility: y-values oscillate and do not approach a single value Example: . NEVER SETTLES DOWN! Limit as x->infinity DNE Y-values keep taking on every value between -1 and 1 forever.

Infinite Limits Definition: “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: “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. IN THE ABOVE SITUATIONS, THE ONLY LIMIT AT INFINITY THAT EXISITS WAS THE MIDDLE ONE BECAUSE THE Y-VALUES WERE APPROACHING A SINGLE REAL NUMBER. Talk about alternative notations… f->L as x->a …. Read arrows as “approaches”

Calculating Limits at Infinity *The Limit Laws listed previously are still valid if “ ” is replaced by “ ” Limit Laws for Infinite Limits (abbreviated): Also show that –infinity* infinity=-infinity…. Show indeterminate forms… where c is any non-zero constant

Calculating Limits at Infinity Theorem: If r>0 is a rational number, then If r>0 is a rational number such that is defined for all x, then Notice that 1/ large value = close to zero? So … non-zero number / +/- infinity = 0 Really want to have expressions of 1/ infinity because they will go to zero

Calculating Limits at Infinity Examples: Find the limit or show that it does not exist. (a) (b) Infinity / infinity and infinity – infinity are indeterminate ** divide all terms by the highest power of x found in the denominator WHEN YOU ARE COMPARING INFINITY TO ITSELF (DIFFERENCE OR DIVIDING), WE HAVE PROBLEMS… DIFFERENT SIZES OF INFINITY… REALLY ABSTRACT… LAST ONE….x= - sqrt(x^2) when x is less than 0

Horizontal Asymptotes Definition: The line y=L is called a horizontal asymptote of the curve y=f(x) if either Example: Basic functions we know that have HAs: . Recall: we were studying limits of functions “at” infinity (or – infinity)… so we are asking, “what is happening to the y-values as we keep increasing the x-values without a bound?” or what happens to the y-values of the graph as we go really far to the right or to the left?” in the event that they eventually settle down to one number, we say that the limits at infinity EXIST and we call the number that they settle down to a HORIZONTAL ASYMPTOTE of the function Arctanx, e^x, 1/x, 1/x^2

Calculating Limits at Infinity ** The limit of a polynomial as x approaches positive or negative infinity is determined by its highest power ** Example: Determine the end behaviour of . Notice, polynomials will never have horizontal asymptotes but we can still calculate limits at infinity in order to determine end behaviour. limit of f(x) = limit of x^7 and (b) limit of f(x) = limit of 2x^2/x^3 = limit of 2/x = 0 (limit exists so we call y=0 a HA of the function) Note: for (b), we could have also divided each term by the highest power in the denominator but we tried this different approach of selecting the highest power on top to represent the numerator and the highest power in the bottom to represent the denominator to find a simple function with the same end behaviour (limits at infinity) as the given function

Limits at Infinity What about the limits at infinity of these functions? (a) (b) Which part (top or bottom) goes to infinity faster? Difficult to say since infinity/infinity is an “indeterminate form” which means that it doesn’t tell us any information other than we have to do some more work first…. GRAPH!!! We can’t factor like we did with polynomials because we are comparing different types of functions…

Limits at Infinity Difficult to say since infinity/infinity is an “indeterminate form” which means that it doesn’t tell us any information other than we have to do some more work first…. GRAPH!!! GO BACK AND ANSWER QUESTIONS ABOUT LIMITS!!!

Comparing Functions That Approach at Suppose and f(x) approaches infinity faster than g(x) if f(x) approaches infinity slower than g(x) if f(x) and g(x) approach infinity at the same rate if where L is any finite number other than 0. Draw littie diagrams… Finish original two limit questions.

Comparing Functions That Approach at The Basic Functions in Increasing Order of Speed Note: The constant can be any positive number and does not change the order of the functions. Function Comments Goes to infinity slowly with Approaches infinity faster for larger What this table says: Any power function, however small the power n, beats the logarithm on the way to infinity. Any exponential function with a positive parameter beta in the exponent beats any power function.

Comparing Functions That Approach at

Limits at Infinity What about the limits at infinity of these functions? (a) (b) Which part (top or bottom) goes to 0 faster? Graph! Semi-log graph.

Limits at Infinity Semilog Graphs Graph! Semi-log graph. GO BACK AND ANSWER LIMIT QUESTIONS!!!

Comparing Functions That Approach at Suppose and f(x) approaches 0 faster than g(x) if f(x) approaches 0 slower than g(x) if f(x) and g(x) approach 0 at the same rate if where L is any finite number other than 0. Draw littie diagrams… remember, 0/#=0 and #/0=infinity. The top is already zero while g(x) is still some non zero number so the quotient is zero.

Comparing Functions That Approach at The Basic Functions in Increasing Order of Speed Note: Again, a can be any positive constant and this will not affect the ordering. Function Comments with Approaches 0 faster for larger Approaches 0 really fast PROOF: L’Hopital’s Rule Remember: only true for very large values of x!!!

Comparing Functions That Approach at PROOF: L’Hopital’s Rule Remember: only true for very large values of x!!!

Comparing Functions That Approach at Semilog Graphs Here, its much easier to see which function is decreasing the fastest…. Easiest to see the speed at which the functions are decreasing.