Basic Math Formulas. Unit circle. and. Arithmetic operations (ab means a b) Powers and roots. a(b + c)= ab + ac

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1 Basic Math Formulas Arithmetic operations (ab means ab) Powers and roots a(b + c)= ab + ac a+b c = a b c + c a b + c d = ad+bc bd a b = a c d b d c a c = ac b d bd a b = a+b ( a ) b = ab (y) a = a y a a = a n n = m n n = m a = a b b n y ( y )a = a y b n n = y Logarithm: loga = y a y = Natural logarithm: ln = loge (e,788..) loga (y) = loga () + loga (y) loga (/y) = loga () - loga (y) loga ( r ) = r loga () ln () = y e y = ln () = 0 ln(e ) = ln (e) = e ln () = loga () = ln () ln(a) Factors: ( + a)( + b) = + (a + b) + ab ( ± a) = ± a + a ( + a)( a) = - a a +b +c = 0 =>, = b ± b 4ac a Absolute Value (for a > 0): = a => = a or = -a < a => -a < < a > a => > a or < -a Right triangle Pythagorean Theorem: a + b = c sin θ = opp tan θ = hyp = b c, sin θ cos θ = opp adj = b a adj cos θ = hyp = a c and Hypothenuse c θ a Adjacent side b Opposite side Important angles θ θ rad = 80 π sin θ cos θ sin θ tan θ = cos θ 0º º π/6 ½ ½ ⅓ 45º ¼π ½ ½ 60º ⅓π ½ ½ 90º ½π 0 -- Trigonometric properties sin () + cos () = sin(-) = -sin() cos(-) = cos() tan(-) = -tan() sin(½π-) = cos() cos (½π-) = sin() sin( + y) = sin()cos(y) + cos()sin(y) cos( + y) = cos()cos(y) - sin()sin(y) sin() = sin()cos() cos() = -sin () tan() = sin() = cos () - cos() Unit circle θ sin θ cos θ

2 Lines: slope of a line through two points (, y) and (, y) is a = y y The slope (=a)/intercept(=b) form: y = a + b The slope/point (, y) form: y y= a( ) Two lines y = a + b and y = a + b are - parallel if a = a and - perpendicular if a = a The distance between two points P(, y) and P (, y): P P = ( ) +(y y ) Parabolas y = a + b + c (a 0) or y = a ( - d) + e etreme point (d, e) a > 0: parabola opens upward. a < 0: parabola opens downward. y-intercept: y = c -intercept(s): solutions of a + b + c = 0. D = b -4ac > 0 => -intercepts, D = 0 => and D < 0 => 0 -intercepts. Interval where y < 0: solve a + b + c < 0 Functions Properties of a function f() and its graph y = f(). D f = Domain f: all permitted values of. Range f: all possible values of f() ( in D f ). Vertical line test for a function f: = a intersects the graph of f at most once. Even function: f( ) = f(). The graph can be reflected about the y-ais. Odd: f( ) = f(). The graph of the function can be reflected about the origin. 4. Increasing function: > => f ( ) > f( ) Decreasing : > => f ( ) < f( ) 5. Composite function: f g() = f(g()) ( apply f after applying g to ) 6. Periodic function: f has a period c if c is the smallest number such that f(+c) = f(). 7. One-to-one function: => f ( ) f( ) a horizontal line intersects at most once 8. The inverse f - of a one-to-one function f: f - (y) = f() = y Find the inverse by:. Checking if f is one-to-one. solving from y = f() and. Interchanging and y. Transformation of functions:. Shifting of a graph (c > 0) y = f() + c : shift y = f() c units upward y = f() - c : shift c units downward y = f( c): shift c units to the right y = f( + c): shift c units to the left. Stretching and reflecting of a graph (for c >) y = cf(): stretch the graph y = f() vertically by a factor c y = f(): compress vertically by factor c c y = f(c): compress horizontally by factor c y = f(/c): stretch horizontally by factor c

3 y = f(): reflect about the -ais y = f( ): reflect about the y-ais. Graphing the inverse function f - : Reflect the graph y = f() about the line y =. Functions - Important functions Polynomial function of degree n N: y = a n n a + a 0, if an 0 n = 0 constant function: the graph is a horizontal line y = a 0 e.g. y = - n = linear function: line y = a + a 0 e.g: y = + n = quadratic function: the graph is a parabola y = a + a + a 0 e.g. 5 p( ) n = : cubic function, e.g.: p( ) 5 Power functions: f() = a, a R.. a is a positive integer

4 f ( ) f ( ) f ( ) 4. a is a negative integer e.g. a = and a = : f ( ) f ( ). a = k n is a rational number: f ( ) f ( ) 4

5 Rational functions:f() = P() Q(), for polynomials P() and Q(). Domain: Q() 0! f( ) 5 6 Algebraic functions: a function that is constructed from polynomials by algebraic operations: addition, subtraction, multiplication, division or taking roots. e.g: 5 f ( ) 4 4 Eponential functions f() = a, base a > 0 Domain = R, Range = (0, ) if a f( ) 4 f( ) f( ) Logarithmic functions f()= log a (), for a = e: f()= ln () f ( ) log f ( ) log0 f ( ) log 5

6 Trigonometric Functions f() = sin() f() = cos() f() = tan() Odd function Period: π Range:[-, ] Even function Period: π Range:[-, ] Odd function Period: π Range: R Cosecant, secant and cotangent function f ( ) csc sin f ( ) sec cos f ( ) cot tan Note: sin sin ( sin is the inverse of sin ) f() = tan() has period π: f() = tan() is one-to-one if Df = (-½π, ½π): Then the inverse function eists on D f : f - () = tan - () 6

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8 Eercises Calculus: Basic Math and Functions Some eercises to improve your basic math skills (James Stewart, Appendices A, B and D) A Rewrite without absolute value: a. 5-5 b. + c. + A Solve the inequality and give the solution set as an interval: a. + 7 > b. c. + < 5-8 d. 4 < + + e. - 0 A Solve: a. = b. + 5 f. < 4 A4 Solve for negative constants a, b and c: a + b < c A5 Solve, if possible, by factorizing: a = 0 b. = c = 0 d. = -0 A6 If a ball is kicked upward from the top of a building 8 ft high with initial velocity 6 ft/s, then the height h above the ground t seconds later will be h = 8 + 6t 6 t. During what time interval will the ball be at least ft above ground? B Find the distance between the points (, ) and (4, 5) B Find the slope of the line through the points P(-, ) and Q (-, -6) B Find the equation of the line if a. Through (,-) and slope 6 b. the -intercept = and y-intercept = - c. Through (-, -), perpendicular to the line + 5y =8 D Find sin θ, cos θ and tan θ for θ = a. π/4 b. 9π/ c. 5π/6 D Find cos θ and tan θ if sin θ = /5, 0 < θ < ½π D If in a right angle triangle the hypotenuse has length 0 and θ = 0 0, compute the length of the adjacent side D4 Find all values of in [0, π] satisfying: a. cos - = 0 b. sin = c. sin < ½ d. sin(½) = ½ Test your basic math skills (compute the result yourself and choose the correct answer a, b, c or d) a b c d 0. = Simplify Simplify (a + ) (a+) = a + 5 0a = = If a = and b = -, 7. then -(a b) - (a b ) = 8. Solve or y = 8 9. Solve y from { -/ / y = 5 0. Solve cos () = ½ for ½π π 5 π, 7 π 5 π, 7 π π, 4 π π, 5 π cos = sin - cos sin - cos cos -. sin (½π + ) = - sin - cos cos sin. The number of -intercepts of f() = is ln(e - e ) = ln() ln(4.5) + ln(e - ) ln(e - ) 5. The solution of the equation e = 9 is ln ln 4.5 (ln ) / ln 8 6. Which of the equalities is/are correct? (more than answer is possible) p pq p q p pq p(p+q) p pq p p pq q p q Which of the equalities is/are correct for all 0,, -? Which of the equalities is /are correct for p > 0 Write 5+ as a b (where a and b are the smallest possible positive integers) + = + = + + p 8 = ( p) 8 p 8 8 = p Answer: + = + p 8 = (p 8) + = 8

9 0. If c = 00 R, epress R in c and m Answer: m+ R Eercises for Functions and their properties. Find the domain of the function a. f() = + b. f(t) = t. Find the domain and the range and sketch the graph of the function + if - b. f()= { a. g() = 5 if > -. Determine whether f is even, odd or neither. If f is even or odd use symmetry to sketch its graph a. f() = - b. f() = + c. f() = -.4 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, eponential or logarithmic function: a. f() = b. g() = c. h() = d. k() = + + e. r() = tan f. s() = log0.5 Suppose the graph of f is given (e.g. f() = ).Write equations for the graph that are obtained from the graph y = f() as follows: a. Shift units upward b. Shift units downward c. Shift units to the right d. Shift units to the left. e. Reflect about the -ais. f. Reflect about the y-ais g. Stretch vertically by a factor of h. Shrink vertically by a factor..6 Given is the graph y =. Sketch it and also a. y = f( + ) b. y = f() + c. y = f() d. y = ½ f() +.7 Graph each function by starting with the graph of a standard function and then applying an appropriate transformation: a. y = -/ b. y = cos (/) c. y = d. y = +.8 Find f+g, f-g, fg and f/g if f()= + and g() =.9 Find f g, g f, f f and g g and state its domain if a. f()= - and g() = + b. f() = and g() = c. f() = +.0 Find f g h and state its domain if f() =, g() = and h() =. Write an equation that defines the eponential function with base a > 0. State the domain and range if a. Sketch the general shape of the eponential function for the cases: () a > () a = () 0 < a <. Use the general shapes of. to give a rough sketch of a. y = + b. y = c. y =. Find the eponential function f() = Ca whose graph is through the points (,6) and (, 4) Eercises inverse functions and logarithmic functions:.4 Find the eact value of each epression: a. log (64) b. log 6 ( ) c. log ()+log (5) ln () d. e 6.5 Epress the given quantity as a single logarithm: a. ln(4) - ln() b. ln () +a ln (y) b ln(z).6 Solve each equation for : a. e = 6 b. ln() = - c. 5 = d. ln() ln(-) =.7 a. What is a one-to-one function? b. How can you tell from the graph that the inverse function eists?.8 Determine whether f() is one-to-one if a. f() = 7 - b. g() =.9 Find the formula of the inverse function if a. f() = + b. f() = + 5 c. f() = ln( + ).0 If a bacteria population starts with 00 bacteria and doubles every three hours then the number of bacteria after t hours is n = f(t) = 00 t/ a. Find the inverse of this function and eplain its meaning. b. When will the population reach 50000? 5 9

10 Solutions to the etra basic math eercises: A a. 5-5 b. + for - - for < c. + a. > - b. - c. > d. - < ½ e. f. < 0 or > ¼ a. = ±/ b. -7 or - 4. > (c-b)/a 5a. 5 5b., -9 B / a. y = 6-5 b. y = - c. 5 y + = 0 D a. sin(¾π) = ½, cos(¾π) = -½, tan(¾π) = - b. sin(9π/) =, cos(9π/) = 0, tan(9π/): undefined c. ½, ½, ⅓. cos θ = 4/5, tan θ = ¾. 5 4a. ⅓π, 5π/ 4b. ¼ π, ¾π, 5π/4, π/ 5c. ⅓ 5d. none (D = 9-40< 0) 6. [0, ] or 0 t (t > 0!) 4c. 0 π/6 and 5 π/6 π 4d. ⅔π, 4π/ Answers to the test: d b c a d b b c c a d c b c a c b, c, d b, c ( mc 00 c ) if mc 00 c > 0 Solutions for the Function-eercises:. a. D = { ±} b. D = (-, ). a. D = [5, ), R = [0, ) b. D = (-, ), R = (-, ). a. Even (graph by reflecting about the y-ais) b. Neither c. Odd (graph by reflecting about the origin).4 a. root b. algebraic c. polynomial, degree 9 d. rational e. trigonometric e. logarithmic.5 a. y = f() + b. y = f() - c. y = f( - ) d. y = f( + ) e. y =- f() f. y = f(-) g. y = f() h. y = ⅓f().7 a. Start with y = / and reflect it about the -ais b. Start with y = cos and stretch horizontally by a factor c. Start with y = / and shift it to the right units. d. y = + = - ( +) = ( - ), so: start with y =, shift it to the right, reflect about the -ais and shift it upward..8 f+g ()= + +, f-g ()= - + +, fg() = ( + )( ) and f/g() = +.9 a. f g() = (+) (+), g f()=( ) +, f f()= ( ) ( ) and g g() = (D= (-, ) for all) b. f g() = (D = [, ) ), g f() = - (D = [, ) ), f f()= (D = [, ) ), and g g() = 4 (D = (-, ) ) c. f g() = (D = { 0}), g f() = (D = { 0}), f f() = (D = { 0}) and g g() = ( + ) + ( + ) (D = R).0 f g h() = (D = [, ) ). f() = a, D = (-, ), R = (0, ) (the graph is increasing for a >, y = for a = and decreasing for 0 < a <. a. Graph y = (it`s an increasing graph) and shift unit upward b. y = = ( ) (decreasing graph) c. Graph y =, reflect it about the -ais and shift it units upward (decreasing graph, y-intercept = ). a =, C =.4 a. 6 b. - c. 5 d. 8.5 a. ln(8) = ln() b. ln ( ya ) z b.6 a. 4 ln() b. c. 5 + log e () = 5+ ln()/ln() ( > because of the domain of ln(-)!) d. e e.7 a. If implies f ( ) f( ) b. Every horizontal line intersects the graph at most once..8 a. Yes b. No.9 a. f () = 5 b. f () = ( 0) c. f () = e a. f - ( n) = log (n/00) = ln(n/00)/ln() b. After about 6.9 hours (f - (50000)) 0