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UTM / MAT134

1500

If then Corollary ()

2200

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

Indefinite Integrals Given let and if is an anti derivative of then :

1100

5, 10, 15, 20, 25 ... {} where ... ... {} in general An infinite sequence is a function where domain is the set of positive integers and range is a set of real numbers As increases the term of a sequence gets close to a real number If a sequence {} has a limit it is convergent, if not its divergent only if therefore Sequence is increasing if If the sequence is bounded and monotonic (neither increasing or decreasing) examples

How to evaluate Case 1 Both and are EVEN Use these identities: & Example Take out Take out Integrate Case 2 When is odd Use identity and substitute Example Split Expand Case 3 When is ODD Use identity and substitute Example How to evaluate: or Case 1 When is EVEN and is ODD Use identity Example Case 2 When and are ODD Use

2400

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

800

{} is an infinite sequences ... The partial sum is If therefore infinite series is convergent where = sum diverges if or it fails to exist Divergence Test If or DNE then is divergent Converges to this if whereas if series diverges

2900

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 /

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* 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

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Sign=ma Notation If am ,am+1.....an are real numbers and m & n are any positive integer such that m ≤ n then : = .... Algebra rules for finite sums Sum/difference : Constant multiple rule : Constant value rule : Sum of the first n integers Sum of the first n squares : Sum of the first n cubes : Reimann sum Let be a function defined on the integral [ a,b} if P is a partition of [ a, b ] and S is a selection for p , then the Reimann sum for by P & S is : R =

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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

Type 1: Infinite Integrals If is cont' on then If is con't on then If limit does exist then the integrals CONVERGE if the limit does not exist then integrals DIVERGE Example convergent Type 2: Discontinuous Integrals diverges If is con't on then, OR If is cont; on then,

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

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Arc Length Sum of all the distances Factor out Introduce bounds Change in Change in Example Find the length of for Factorization If f'(x) does not exist, find and then use: Example Find the length of Express in terms of Plug in & to find new bounds

Volume by Disks The volume of this slice is a solid that lies between two planes ⊥ to the -axis at and . If the cross section area of ⊥ to -axis is then volume of is Example Find the volume of a solid with the unit disk as its base and its cross sections perpendicular to the -axis are right isosceles triangle with one leg in the disk. Solution Since the isosceles the base and height are equal So how to find base ? At point so, Now we know: Ex

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Definite Integral The definite integral of f (x) from a to b is denoted by : where K E R If on [a , b ] and then , = area If on [a , b} then , if c is b/w a & b If on [ a, b } then, Average value of a function if is integratable on [ a ,b ] > Average value Ȳ of for x in [a , b ] is

1000

Ratio Test if (absolutely convergent) (divergent) (divergent) (in-determined) Root Test if (convergent) (divergent) (in-determined)

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

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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 .

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Theorem of Calculus Let f be continuous on [a,b] . If a≤ c ≤ b and F is any antiderivative of f then for all x E [a,b]