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Properties of Inequalities Notation We write for “ is less than ” We write for or Facts (AP – 1 ) have the same sign and Ex. Ex. Definition: The Real Number Line The statement means that is on the left of in the number line. We give example of solving inequalities. Ex : Solve Ex : Solve *Note that the left hand side is positive. Thus, We get We obtain: We get: Ex : Solve We introduced to set notation. Note: som

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Real Number - Can be represented as a decimal Integers - ,, , , ... Non Negative Numbers - , , , ... Natural Numbers - , , Rational Numbers - where & are integers and Irrational Numbers - Cannot be represented as a rational (ex. , , ) Interval - a set of all real numbers can be represented as a line on the number line [include] (exclude) Number Lines Absolute Value - denoted by , defined by formula : triangle equality Solving Inequality thus we want

We introduce the notion and notation for functions of a single real variable. Idea: A function is rule that associates a single output to any input Synonyms Algorithm Rule Mapping Transformation Example The area of a square is a function of its side length The volume of a sphere is a function of its radius Notation : means that is a FUNCTION of . Here represents the function. We call the INDEPENDENT VARIABLE and the DEPENDENT VARIABLE. Definition: The set of all admissi

Function - A relationship between two variables Vertical Line Test Graphs Liner Functions → y= mx + b form -Parallel slopes are equal [ m1 = m2 ] -Perpendicular when m1 = -1/m2 2. f(x) = y = √x → D = [ 0, ∞] , R = [0 , ∞] 3. f(x) = y= x3 → D= R =(-∞, ∞) , R= R (-∞, ∞) 4. Piece wise Function → f(x) ={x² ,x < 0 ; 1 , 0 ≤ x < 3 ; -x ≤ x } D = R

Algebra Functions 1. Sum Differences Ex 2. Product & Ex 3. Quotient of & Ex and Composition Of Functions The domain of the consist of numbers in the domain of for which lies in domain of Shifting and Scaling Graphs Shift if , shift ↓ if Shift right if , left if Stretches vertically Compresses vertically Compresses horizontally Stretches horizontally Reflection on axis Reflection on axis

Transforming Graphs Question: How does the graph of relate to the graph of ? Consider, We see that the graph is SHIFTED to the left by three units. Question: How does the graph of relate to the graph of ? Consider, In general , the graph of is the graph of shifted: To the left if is positive To the right if is negative Vertical Shifts Question: How does relate to ? Consider the example, The graph is vertically shifted up one unit. Question: How does relate to ? Con

where is in radians Conversion: radians deg. radians deg. deg radians degs. radians Special Triangles Other Basic Rules Identities from 2.4

Trigonometry Fundamentally , trigonometry is about the unit circle in the plane. The unit circle gives: For general triangle, we have: (co-secant) (secant) (co-tangent) All angles will be measured in radians. Definition: An Angle of RADIANS measures an arc of units on the unit circle. Thus , radians degrees In general, The Special Triangles We obtain: We calculate: We obtain: Example: Calculate and We obtain Graphing Trig Functions The g

* cannot be negative because if is a fraction, you can't root a negative # ^if if if if or *if and applies to all real numbers & Exponential Function Power Function Interest Example: In the year 2000, you invest $100 into a savings account w/ 5.5% interest compound annually. Find the formula for amount of $ in years. Natural Exponential Functions where ... (an irrational #) & Exponential Growth & Decay where is a constant and if = growth, = dec

Exponentials Definition An exponential function has the form for and The following exponent rules govern the behaviour of exponential functions: Laws Example - Simplify Notation There is a special base called Euler's Number given which is often used as the base of exponentials. Logarithms Definition The Logarithm Function (with base a) is defined for and such that To put it plainly, is the exponent we must raise to, in order to obtain . Example si

Definition A function is ONE-TO-ONE on a domain if Whenever in Plainly is one-to-one if every gets sent to a different value Definition The Horizontal Line Test - A function is one-to-one if and only if each horizontal line meets it graph at most once. Discuss Which of these functions is one-to-one ? Definition If is one-to-one on and has range then the INVERSE FUNCTION is defined by: if Why do we need one-to-one? If then what is ? It could be or

One-to-One Function A function f is on domain if where in (i.e one for every only) Horizontal Line Test NOT one-to-one If has horizontal line going through point its one-to-one. Specifying domain can help! One-to-one function Inverse Function Inverse if is one-to-one then its inverse function is something that satisfies if on graph on graph So, of is of & of is of To find inverse just switch & and solve for Logarithmic Functi

The Idea Of Limits Limits are a tool used in calculus to learn about the value of a function at by studying the values near Example Complete the following tables of Values: It looks like is close to and so we expect Near the function is near Definition If is arbitrarily close to when is sufficiently close to then THE LIMIT OF AS APPROACHES IS and we write Mainly, " is close to when is close to " Example Calculate Consider If is close to then is c

Average rate of change of on now on If gets closer and closer to a number as gets closer to from both sides, then is the limit of as approaches . Example Use a table to estimate the limit: Obviously as the denominator As approaches the limit seems to be Graphically: When holes or jumps occur in a function curve the graph is not continuous. Limits can be displayed in various ways: As These functions do not have a as The Limit Laws , , and are re

To have a limit as approaches , the function must be defined on both sides of c and the values must approach Two sides limits Functions can have one sided limit if approach from one side A right hand limit A left hand limit Rule The limit exits if a function has a limit as approaches . If it has left handed and right handed limits and both are equal If Rule Also remember this side rule to help solve problem , where can be replaced by Example 1 Does the l

Limits We briefly recall the notion of a limit. is close to when is close to Example Calculate Because the limit is a simple polynomial we may evaluate by substitution. Example Calculate We need to be careful with direct evaluation. Example Calculate and Thus we get "" which is not a number One Sided Limits Discuss What is the behaviour of "close to" ? Consider the graph: Sloping up when Not define

Continuity of a Point At function does not have a limit Not continuous at At but Not continuous at At but not continuous at is continuous at is: But, At and Continuous at At and Continuous at Continuity Test A function is continuous at if: exists exists If one of these fail the function is discontinuous Types of discontinuity Jump Infinity Oscillating Removed point Theorems Polynomial function is continuous at every number

Suppose a patients blood sugar level looks like : What could the events at hours and hours mean ? Jumps in a graph signify something interesting Continuity Test - is CONTINUOUS at if : exists exists EXample Is the function continuous at ? exists Thus does not exist and is not continuous at We can now reframe the limit laws as laws about continuity. Facts If and are continuous at then the following are also continuous at

Limits Remember where where K = constant Horizontal Asymptote The line y=L is H.A of the curve if or Remember this! 2. no. HA 3. therefore HA @ 4. & 5. & 6. therefore & , therefore Vertical Asymptote The line is a VA of the curve if or

Infinity is not a number. We write the symbol but it is not a number in the usual sense. Never use it in calculations. Vertical Asymptotes Definition We say has limit as as approaches infinity and write if is close to when is large Example We make the table of Values We observe , when is large is small Discuss What happens to arctan () when gets large ? & Horizontal Asymptotes Definition A line is a horizontal asymptote of if: or example Find

Example Find the slope of the curve at any point , find the slope at / / / = = plug in 0 = Therefore at the slope is Slope at an instant p is derivative of a function /

Horizontal Asymptotes Recall, is a HORIZONTAL ASYMPTOTE of if or Example Find the horizontal asymptotes of We find the limits at infinity Cancel Thus, is a horizontal asymptote is tricky. It can equal any finite value EXample Find the horizontal asymptotes of Cancel Visually, the graph flattens EXample F

Rate Of Change Consider the following You are driving at 120 km/hr. What does this mean? Discuss How far do you travel in 2hr? 1hr? 30 mins.? 1 min.? Two photos taken 1 sec. apart show that a car has moved 29m. How fast? Define The AVERAGE RATE OF CHANGE of between and is Discuss What is the average rate of change of from to ? ? Speed Rate of change of distance Instant Rate of Change Example An apple falls with displacement , how fast is it travelling at ?

Notations to indicate the value of a derivative at a specific number Non differentiate functions and their graphs a) at → A sharp turn or point slope is undefined b) at 0 → There's a "cusp" slope = undefined c) at → Vertical tangent line slope = ∞ therefore undefined d) at → Non continuous graph is undefined Theorem - If f is differentiable at therefore f is continuous at ,but the converse is not always true

Derivative as Function Recall from last lecture, The SLOPE of at is This function of is ''derived ' from and we call it the DERIVATIVE of Example Stretch the derivative of Example Calculate if Notation If we write: '"Leibnitz" "Newton" Example Find if Clear denominators Example Find if Conjugate the top One Sided DEfinition

The chain rule The Product Rule The Quotient Rule The Derivative of exponentials/logarithmic functions → if Trigonometric Derivative Basic Rule ,

Recall, "The derivative of at " "The slope of at " Example Find the slope of at Apply the definition of slope Thus, the slope at is Rate of Change The physical interpretation of derivatives is that they measure rates of change. Definition If the distance to an object at time is then its: VELOCITY SPEED Speed ignores direction ACCELERATION Discuss An object dropped near Earth travels meters in seconds. How far do you

Displacement - Average velocity - Vav = Displacement/ time travel = Instant velocity- Speed - Acceleration -

Fact: If then Fact: If then Example Find if Addition Scaling Power Example Find if Example Find if Re-write with negative exponents Example Find all the points where has a horizontal tangent line. Horizontal tangent Thus, there are horizontal tangent lines at: and Evaluate for Thus, the tangents at and a

Differentiation Rules Recall, We introduce more functions Fact: - Is its own derivative Example Find if Product rule Example Find if Quotient Rule The constant can be defined by Higher Order Derivatives The SECOND DERIVATIVE of is Acceleration is the derivative of speed and the second derivative of position In general, the nth DERIVATIVE is: Example Find and of Calculate the first derivative Calculate the second d

Implicit Differentiation therefore This is what we want so to isolate for this Is the slope of the tangent line. therefore

The Chain Rule Question: How does relate to and ? Example Suppose there are 12 hockey cards in a pack and there are 20 packs in a case. How many hockey cards are there in five cases? Number of cards (12 cards/pack) (20 packs/case)(5cases) Example Suppose 1 kg of beans costs 2 $ and each kg has 1000 beans. How many beans do you get for 6$? Number of beans Fundamental Idea Compound rates of change are obtained by multiplying their parts Example If & Find

Example - snow proof Taking the natural logarithm to solve problems Example Find if Solution apply "In" to both sides cos x comes down now derive .

So far, we have covered functions with explicit formulas: Sometimes we do not get a formula but instead we get a relation. We cannot solve for in this equation Example Find if Assume for some unknown function Differentiate the relation Replace and Solve for Example Find the slope of the tangent line to At the point Differentiate both the sides Evaluate for the sign Discuss The fundamental law of muscle contraction

Derivatives and Inverses Recall, the inverse of a function is a function such that: EXAMPLE Find the inverse of Solve for in terms of Switch and Define Check: Thus, is the inverse of . The notation is very common. QUESTION If then how are and related? (HINT: Use the chain rule) FACT DISCUSS If find Recall, KEY FACT EXAMPLE Find and DISCUSS Calculate using a calculator and at and compare with EXAMPLE Suppose

Trigonometric Graphs is differentiable throughout the intervals but not at because of vertical tangents

Recall, These are the inverse of standard trig functions. OR and OR and Discuss Where do the restrictions on in the definitions of and come from? Answer and thus and thus What is the slope of arctan? Example Calculate (The famous Formula Thirteen) Find the derivative of Apply the inverse formula Find Construct a triangle with unit hypothenuse that satisfies Thus,

Extreme Values A function has an absolute (global) maximum at a if for all domain of . Similarly has a global minimum at if 2. A function has a relative (local) maximum at a if there is an open interval containing such that in . Similarly has a local maximum at there is an interval containing for all in . 3. Extreme value theorem If is continuous on a closed interval the attains an absolute max and minimum in , to find absolute max and min of a continuous function on a c

We now introduce the primary method by which calculus enters in to applications. Recall, the technique of implicit differentiation. To find the derivative given Differentiate both sides Isolate for Observe - To understand we only need a relationship among the variables and . We do not need an explicit function . Ex. A kite is 10m off the ground and 20m to the right. If the wind is blowing it at 2m/s to the right then how fast is the string unwinding? Draw a picture Define the va

A function f is increasing if A function f is decreasing if A function f is monotonic if it is either increasing or decreasing Rolles's Theorem If a function f is : Continuous on a closed interval [a , b] Differentiable on an open interval (a , b) then for atleast one number c in ( a , b ) Example - Show that is not differrentiable on { 0 , 2 } Solution- so if if so is continuous on [ 0 , 2 ] suppose is differentiable on ( 0 , 2 ) (according to the

Extreme Values We have the detailed study of the structure of graphs of functions. Flow are they shaped? What are their key properties? Definition → Let a function be defined on the domain . is an ABSOLUTE MAXIMUM if → for all in is an ABSOLUTE MINIMUM if → for all in We say there are EXTREME VALUES of EXAMPLE Find the global min of on We know thus is the minimum Example If possible find the global min/max of on the domain We have is a minimum.However, is not in the domain a

Mean Value We begin our study of how the derivative of a function effects its graph. Thm → (Rolle) If is differentiable on and then there is in so that - Idea → Throw a ball up and catch it at the some height. At same time in between the ball must be stationary. Example Find the so that # Calculate the derivative # Solve Thus, is the solution given by Rolle's theorem Example Show that there is one root of in the interval Without using the quadratic equation #Apply

Testing for monotonic functions if then f is ↑ If then f is ↓ If then f is constant The first deriviative test : for local extrema "c" is a critical number of a function f that is continuous on [a , b ] then : if for and for = local max at c 2. if for and for = local min at c 3. if does not change sign at c therefore no local extrema

Point of reflection - curve changes from one concavity to another Curve sketching critical values from first derivative Example at x = 0 , -2 critical points 2. Intervals ↑ or ↓ around CP 3. Local extrema values 4. Concavity using second derivative = = +ve up! = - ve down ! = +ve up! 5. Sketch Second derivative test & local min at c & local max at c

Monotonic Functions monotonic mono tone This week we build up this tools required to accurately sketch graphs and understand the structure of functions. Define is increasing on if - in is decreasing on if - in Example is increasing on If then . Thus, is increasingly on is decreasing on if then and are negative. Thus and . We get Therefore, /x/ is decreasing on INC/DEC and Slopes Fact → If on then is increasing. If on then is decreasing. By

Concavity Define Let be differentiable. is concave up if is increasing concave down if is decreasing Concave up Concave down Example The curve is concave up The curve is concave down Discuss Determine where is concave up and concave down Differentiate f(x) Determine where is increasing/decreasing Thus, f(x) is concave up on and concave down on The Second Derivative Test for Concavity If the f(x) is concave up. If then f(x) is concave down. Example Find the interva

Indeterminate Forms The function is indeterminate if it is not definitely or precisely determinate There are 7 interminate forms % 00 ∞0 ∞ L" Hopital's Rule if , then differentiate both & then apply the limit . Thus Example But IIm

Curve Sketching Recall Critical points or undef Increasing Decreasing Concave up Concave down Point of inflection 1) Tangent line 2) Concavity change The Curve Sketching Algorithm Identify the domain, symmetries, intercepts Calculate and Find the critical points and their behaviour Increasing/decreasing Concavity and points of inflection Asymptoles Draw the curve example Sketch 1) Domain y-intercept x-intercept odd symmetry 2) 3) If then