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Calculus 1, Exam 1 Study Guide

by: Monica Chang

Calculus 1, Exam 1 Study Guide 21-111

Marketplace > Carnegie Mellon University > Mathematical Sciences > 21-111 > Calculus 1 Exam 1 Study Guide
Monica Chang

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Material from weeks 1-6 - Basics review - Functions and models
Calculus 1
Deborah Brandon
Study Guide
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This 15 page Study Guide was uploaded by Monica Chang on Monday October 3, 2016. The Study Guide belongs to 21-111 at Carnegie Mellon University taught by Deborah Brandon in Fall 2016. Since its upload, it has received 38 views. For similar materials see Calculus 1 in Mathematical Sciences at Carnegie Mellon University.


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Date Created: 10/03/16
21-111 Calculus 1 Study Guide Arithmetic:  a+b=b+a  a+b +c=a+(b+c)  ab+c =ab+ac  ab=ba  ab)c=a(bc) Multiplying Fractions:  a× = ac b d bd Dividing Fractions: a c ad  b÷ d bc Adding Fractions: a c ad cb ad+bc  + = + = b d bd bd bd ExponentnBasics:  a means multiply a by itself n times n  0 =0wheren>0  a =1wherea≠0 0  0 isindeterminate 1  a = nwherea≠0 a because 0 n−n n −n 1=an=−n =a a 1=a a 1 −n n=a a Laws of Exponents: Given m & n are integers and a & b are real numbers: m n m+n  a a =a am m−n  n =a wherea≠0 a a a  (¿¿n)m n mn (¿¿ m) =a =¿ ¿  (ab) =a bn n n ( ) = a whereb≠0  b bn −n n n a b b  ( ) =( ) = n b a a Root Basics: 1  √a=a n n m n m  √a =( √) 3 o Ex. Calculate 82 . 3 2 3 2 We would rather do (√8) than do √8 . Root Laws: 1 1 1  n n n n n n √ab=(ab) =a b = a √ √ 1 1 n na a n an √a  =( ) = 1= n √b b n √b b Factoring Formulas:  x −y =(x−y)(x+y) 2 2 2  x +2 xy+y =(x+y) 2 2 2  x −2xy+y =(x−y) 3 3 2 2  x −y =(x−y)(x +xy+y )  3 3 2 2 x +y =(x+y)(x −xy+y ) You can expand the following equations easily using Pascal’s Triangle to 0ind the coefficients of each term:  (x+y) =1  (x+y) =x+y 2 2 2  (x+y) =x +2 xy+y  (x+y) =x +3x y+3 x y +y Here are the starting rows of Pascal’s Triangle: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 … Rationalizing: 2 2  x −y =(x−y)(x+y) We call (x−y) and (x+y) conjugates. o Ex. Rationalize √ a−√b . √ √+ b) a−b √a− √= √ a−√b × = √ √+ b) √ √ b Note: In calculus, we do not care about having radical signs in the denominator. Completing the Square:  Use the following idea: x +2 xy+y =(x+y)2 2 2 2 x +2 xy=(x+y) −y o Ex1. Solve for x: x +8x+3=0 . 2 x +8x+3=0 x +8x=­3 x +8x+16=­3+16 2 (x+4) =13 x+4=± 1√ x=± 1√−4 2 o Ex2. Complete the square: F x =x −7x+3 F x)=x −7x+3 2 2 F x)=x +2 −7 x+ −7 − −7 +3 ( 2 ( ) (2) 2 2 2 F x)=(x− ) +3− −7 2 ( 2 2 F x)=(x− ) − 37 2 4 Writing an equation in this form can make it so much easier to graph Quadratic Formula: 2  For equations in the form a x +bx+c=0 , you can use also the quadratic formula to solve for x: 2 x= −b± √ −4ac 2a Methods:  Now you know 3 methods to solving equations in the 2 form a x +bx+c=0 . o Factoring o Completing the Square o Quadratic Formula Long Division:  Example 1: 4 x −4x +x−1Remainder6 4 3 2 x+1 √x +0x −3x +0 x+5 −(4 x +4 x ) 3 2 −4x −3x −(−4x −4x )2 x +0 x 2 −(x +x) −x+5 −(−x−1) 6 4x −3x +5 3 2 6 So =4x −4x +x−1+ x+1 x+1  Example 2: Factor 3 2 x −2 x −5x+6 o Use trial and error to find a root x so that x −2 x −5x+6=0 o x=3 is a root, so x-3 is a factor x −2x −5x+6 2 o use long division to find that x−3 =x +x−2 3 2 2 o x −2 x −5x+6= x−( )x +x−2 ) ¿ x−3 )(x−1 (x+2) Intervals & Inequalities:  Simple examples: Interval Notation Number Line Inequality (a, b] a < x ≤ b (-∞, b) x ≤ b  Inequalities in graphing example: Unions & Intersections:  Given two sets A, B o Intersection A ∩ B = all objects in both A and B o Union A ∪ B = all objects in A or B  Example: If A=[−∞,1)∧B= (0,∞) A∩B= 0,1 ) [ A∪B=(−∞,∞) Solving Inequalities:  Ex1. 2−5 x>7 −5x>5 x<−1 Note: you have to flip the sign x∈(−∞,−1)  Ex2. 4<3−x ≤6 4<3−x 3−x ≤6 AND 1<−x AND −x≤3 x<−1 AND x ≥−3 x∈ −∞,−1 ) AND ¿ x∈ −∞,−1 ) ∩¿ x∈¿  Ex32 x −1≥0 (x−1)(x+1)≥0 (x−1≥0∧x+1≥0 ∨ x−1≤0∧x+1≤0 ) (x≥1∧x≥−1 )∨(x≤1∧x≤−1) −∞ ,−1] x∈ [, )∨¿ x∈ −∞ ,−1 ]¿ Absolute value: zif z≥0  ||={ −zif z≤0 ¿thecaseth||≤a (this is an “AND” situation) o if a>0then−a≤z≤a  Logically: if we want the distance from z to 0 to be less than or equal to a when a is greater than 0, z has to be between –a and a, inclusive  Mathematical derivation (only doing this for this case as an example): (z>0∧z≤a ∨(z<0∧z≥−a) z∈(0,a]∨¿ z∈[−a,a] o if a<0then∅no solution  Logically: absolute value cannot be negative o if a=0thenonlysolution isz=0  Logically: if the distance between z and 0 is less than or equal to a and a is 0, z can only be 0 because absolute value cannot be negative ¿thecaseth||≥a (this is an “OR” situation) o if a>0th−∞,−z ∪ z,∞ )  Logically: if we want the distance from z to 0 to be greater than or equal to a when a is greater than 0, z has to include everything except the numbers between –a and a. o if a<0then−∞≤z≤∞  Logically: if we want the distance from z to 0 to be greater than or equal to a when a is less than 0, z can be anything because distance is always a positive value anyway o if a=0then−∞≤ z≤∞  Logically: if we want the distance from z to 0 to be greater than or equal to a when a is 0, z can be anything because distance is always a positive value anyway  Ex1. 2x+2 ≤8 −8≤2 x+2<8 −10≤2x≤6 −5≤ x≤3 x∈¿ 5, 3]  Ex2. 3x|<−8 ?nosolution(becauseabsolutevaluecannot beanegativenumber)  Ex3. 4 x−3>5 4 x−3>5∨4x−3<−5 4 x>8∨4 x<−2 −1 x>2∨x< 2 1 x∈(−∞ ,− )∪ (,∞) 2  Ex4. 2x−4 |>−8 2 x−4>−8∨2 x−4<4 2x>−4∨2x<8 x>−2∨x<4 x∈(−∞,∞) Trigonometry:  Sinθ= opp hyp adj Cosθ= hyp Tanθ= opp = Sinθ adj Cosθ  Pythagorean Theorem: a +B =c 2 2 2 Sin θ+Cos θ=1  There are 2π radians in a circle (360°) Chapter 1: Functions and Models  Function – a rule/pattern we determine from data points to predict values where data is missing (each x value has one y value) x→F →F(x) input functionoutput  A function can only be a function if it passes the vertical line test (i.e. each x value has no more than one y value)  A function is one-to-one if it passes the horizontal line test (i.e. each y value does not have more than one x value)  Ex1. Give an equation, domain, and range for this graph: Equation: x+1,∧x<0 f x)={x ,∧x≥0 Domain: (−∞,∞) Range: −∞,−1 ∪¿  Ex2. Sketch a graph and give the domain & range for this equation: f ( )−2− √−x Graph: Domain: ¿ Range: ¿ 1  Ex3. Find the domain of G(q)=3 √2q−1 Domain = (−∞, )∪( ,∞) 2 2 5  Ex4. Find the domain and range of H(R)= √1−R Domain = (−∞ ,1) Range = (0,∞)  Even, Odd, and Neither functions: o Even function: when f (−x)= f (x)  Symmetric with respect to y-axis o Odd function: when f (−x)=−f (x)  Rotated π radians around the origin o Otherwise it is neither  Ex1. Determine if this function is even or odd: f (x)=x −2 x5 3 5 o f(−x = −x3)−2 5−x ) o f(−x =−x +2 x o f(−x =− (x −2x 5) o f(−x = f(x),so (x)mustbeodd  If f and g are even: o f+g is even o f-g is even o f*g is even o f/g is even  If f and g are odd: o f+g is odd o f-g is odd o f*g is even o f/g is even  If f is even and g is odd: o f+g is neither o f-g is neither o f*g is odd o f/g is odd  Increasing vs. Decreasing functions: o A function is increasing on the interval I if f (1) f( 2whenever x1<x2 in I o A function is decreasing on the interval I if f (1) f( 2whenever x1<x2 in I Linear function:  y=mx+b  To describe a linear function, you need: y −y o m=slope=rateof change= 2 1= ∆ y=tanθ x2−x1 ∆ x o b=y−intercept(wherex=0)  Ex. f (x)=x m=slope=1=tanθ π θ=45°= rad 4 b=0,y− ∫(0,0)  When you want the equation of the family of lines that go through a certain point, you can use point slope form to write the equation: y2−y 1m(x −2 ) 1 Quadratic equation (graphing): o a(x−h )+k o vertex is (h ,) o when a<0 the graph opens downwards a<0 o when the graphs opens upwards o increasing a stretches graph vertically, and vise versa Circle equation: 2 2 2 o (x−h )+(y−k )=r o center is (h,k) o radius is r Power Function: b  In the form y=a x 2 4 4 2  Comparing the graphs of x and x , x is flatter than x when x is between -1 and 1 and steeper for the rest of the domain. 3 5 5 3  Comparing the graphs of x and x , x is flatter than x when x is between -1 and 1 and steeper for the rest of the domain.  Any even root like √ x is neither odd nor even because its domain doesn’t even include –x  Any reciprocal of an even root like 1 is even because √x f (−x)= f (x) 3  Any odd root like √x is an odd function because f (−x)=−f (x) 1  Any reciprocal of an odd function like 3 is odd √ x because f (−x)=−f (x) Polynomials: n n −1  In the form: y = a · n + a n −1· x + … + a · x 1 a 0  The degree of a polynomials is the exponent of its largest exponent term.  The sign of the highest order term will tell you the overall direction of the graph.  If for some value x=p , f (p)=0, we know that x−p must be a factor of the function f (x).  Rational Functions: o f(x)= P(x) where P and Q are polynomials Q(x) o D (f={x∈R , Q(x)≠0 }  Algebraic Functions: o Quotient of “polynomials” where powers can be any real # o Rational functions & polynomial functions are algebraic, but not all algebraic function are rational or polynomial Trigonometric Functions:  In the form: y = a sin (b x + c)  You can use graphs of sin and cos to help with values at different angles. n n  (Cosθ)=cos θ o But note: cos θ≠cosθ n n n  (Sinθ)=sin θ  Sin θ+Cos θ=1  sin−x )=−sin (x)becausesinisanodd function  cos(−x =cos (x)becausecosisaneven function o Example: cos −3x+6π ) ¿cos −3x ) ¿cos(3x) Exponential Function: x  In the form: y=a ,a>0  D (f)=(−∞ ,∞ )  R (f= (0,∞)  b plug∈time x¿ get measureof growth  logbx plug∈growth¿getmeasureof time Logarithmic Function:  In the form: y=log a  D (f)=(0,∞ )  R (f= −∞,∞ ) Graphical transformations: y=− f (x)reflectsy= f (x)about x−axis y= f (−x)reflectsy= f (x)aboutthe y−xis y= f (x)+cshiftsy= f (x)¿theupcunits y= f (x)−cshifts y= f (x)¿thedowncunits y=cf (x)stretches y= f (x)verticallybyc 1 y= ( f (x))compresses y= f (x)verticallybyc c Combinations of functions:  ( f ±g¿ (x= f (x)±g(x)  (fg)x)= f(x)g(x) f(x)  f (x)= provided g(x)≠0 (g g( )  Composite of f & g: (f ∘g)x = f (gx ) o In general, f ∘g≠g∘ f  Ex. f x)=SinX g(x) =CosX (f +g)x)=Sinx+CosX The domain is the intersection of the domains of f and g


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