Algebra/Trig Review Flash Cards. Changes. equation of a line in various forms. quadratic formula. definition of a circle

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1 Math Flash cars Math Flash cars Algebra/Trig Review Flash Cars Changes Formula (Precalculus) Formula (Precalculus) quaratic formula equation of a line in various forms Formula(Precalculus) Definition (Precalculus) Formula for the slope of a line efinition of a circle Formula (Precalculus) Definition (Trigonometry) equation of a circle sin, cos, tan Definition (Trigonometry) Formula (Trigonometry) sec, csc, tan, cot Funamental Trig Ientities

2 Mark Stankus has moife these flash cars to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo. Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License. These cars are part of a project to create free, high quality, printable flash cars for mathematics stuy. You are free to share, moify, an istribute these for non-commercial use, as long as you mention the source URL. Suggestion for aitional material an corrections are welcome. Parts of this set base on L A TEX source from Jason Unerown ( jasonu). Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License. Form Equation point slope y y 1 = m(x x 1 ) slope intercept y = mx + b stanar Ax + By + C = 0 The solutions or roots of the quaratic equation ax 2 + bx + c = 0 are given by x = b ± b 2 4ac 2a A circle with center (h, k) an raius r is the set of points whose istance from the point (h, k) is exactly r. m = y 2 y 1 x 2 x 1 Often the iea of circle an the circle together with its insie are use interchangably. hyp opp θ aj sin θ = opp hyp cos θ = aj hyp tan θ = opp aj The equation of a circle of raius r centere at the point (a, b) is: (x a) 2 + (y b) 2 = r 2 sin 2 x + cos 2 x = 1 sec 2 x tan 2 x = 1 csc 2 x cot 2 x = 1 sec θ = 1 cos θ tan θ = sin θ cos θ csc θ = 1 sin θ cot θ = cos θ sin θ

3 Formula (Trigonometry) Formula (Trigonometry) Double Angle Ientities Half Angle Ientities Formula (Trigonometry) Formula (Trigonometry) Formula with sin(t) an cos(t) Formula with sec(t) an tan(t) Formula (Trigonometry) Definition (Precalculus,Calculus I) Formula with csc(t) an cot(t) Definition of even function Definition (Precalculus,Calculus I) Math Flash cars Definition of o function Flash Cars Math Flash cars Antierivative (Calculus I) Changes x n x (n 1)

4 sin 2 x = 1 (1 cos 2x) 2 cos 2 x = 1 (1 + cos 2x) 2 sin 2x = 2 sin x cos x cos 2x = cos 2 sin 2 x cos 2x = 2 cos 2 x 1 cos 2x = 1 2 sin 2 x sec 2 (t) = 1 + tan 2 (t) tan 2 (t) = sec 2 (t) 1 sin 2 (t) + cos 2 (t) = 1 sin 2 (t) = 1 cos 2 (t) cos 2 (t) = 1 sin 2 (t) f( x) = f(x) for all x csc 2 (t) = 1 + cot 2 (t) cot 2 (t) = csc 2 (t) 1 Examples: x 2, cos(x) These cars are part of a project to create free, high quality, printable flash cars for mathematics stuy. You are free to share, moify, an istribute these for non-commercial use, as long as you mention the source URL. Suggestion for aitional material an corrections are welcome. If you foun these useful, consier contributing back to this project with your time. Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License. f( x) = f(x) Examples: x, sin(x) for all x x n+1 n C Mark Stankus has moife these flash cars to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo. Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License. This only works if n is a constant. We know this from Calculus I if n is a rational number. We will learn this in Calculus II if n is an irrational number, like n = π.

5 Antierivative (7.3) Antierivative (7.3) e x x a x x Antierivative (Calculus I) Antierivative (Calculus I) sin x x cos x x Antierivative (7.2) Antierivative (Calculus I) 1 x x sec 2 x x Antierivative (Calculus I) Antierivative (Calculus I) csc 2 x x sec x tan x x Antierivative (Calculus I) Antierivative (7.2) csc x cot x x tan x x

6 a x ln a + C e x + C This only works for constants a. sin x + C cos x + C tan x + C ln x + C sec x + C cot x + C ln sec x + C csc x + C

7 Antierivative (7.2) Antierivative (7.2) cot x x sec x x Antierivative (7.2) Antierivative (7.6) csc x x x a2 x 2 Antierivative (7.6) Antierivative (7.6) x a 2 + x 2 x x x 2 a 2 First Step (Calculus I) Antierivative (Calculus I) sin 2 (x) x sin 2 (10x) x Antierivative (Calculus I) Vocabulary (Calculus I) cos 2 (75x) x Upper limit of integration in 2 7 x 2 x

8 ln sec x + tan x + C ln sin x + C sin 1 x a + C ln csc x cot x + C This only works for a constant a. 1 a sec 1 x a + C 1 a tan 1 x a + C This only works for constants a. Use Use sin 2 (10x) = 1 2 (1 cos(20x)). sin 2 (x) = 1 2 (1 cos(2x)). 2 Use cos 2 (75x) = 1 2 (1 + cos(150x)).

9 Vocabulary(Calculus I) Lower limit of integration in 2 x 2 x 7 Vocabulary(Calculus I) Integran of 2 x 2 x 7 Vocabulary(Calculus I) Property (Calculus I) Variable of integration of 2 7 x 2 x a a f(x) x Property (Calculus I) Property (Calculus I) a a f(x) x if f is an even function a a f(x) x if f is an o function True or False (Calculus I) Match It! (7.6) b b f(x) x =? f(u) u Domain of arcsin(x) a a Match It! (7.6) Match It! (7.6) Range of arcsin(x) Domain of arccos(x)

10 x x 0 a 2 0 f(x) x This is a great property because plugging in zero is often easier than plugging in a. [ 1, 1] True. Think in terms of graphs. [ 1, 1] [ π 2, π 2 ]

11 Match It! (7.6) Match It! (7.6) Range of arccos(x) Domain of arctan(x) Match It! (7.6) Match It! (7.6) Range of arctan(x) Domain of arcsec(x) Match It! (7.6) Formula (7.6) Range of arcsec(x) x arcsin(x) Formula (7.6) Formula (7.6) x arcsin(u) x arccos(x) Formula (7.6) Formula (7.6) x arccos(u) x arctan(x)

12 (, ) [0, π] (, )?? ( π 2, π 2 ) x arcsin(x) = 1 1 x 2 ( π 2, π 2 ) x arccos(x) = 1 1 x 2 x arcsin(u) = 1 u 1 u 2 x x arctan(x) = x 2 x arccos(u) = 1 u 1 u 2 x

13 Formula (7.6) Formula (7.6) x arctan(u) x arcsec(x) Formula (7.6) x arcsec(u) Graph of y = arcsin(x) Graph of y = arccos(x) Graph of y = arctan(x) Graph of y = arcsec(x) Graph of y = arcsin(x 2) Graph of y = arccos(x 2) Graph of y = arctan(x 2)

14 x arcsec(x) = 1 x x 2 1 x arctan(u) = 1 u 1 + u 2 x x arcsec(u) = 1 u u 2 1 u x

15 Graph of y = arcsec(x 2) Graph of y = 3 arcsin(x) Graph of y = 3 arccos(x) Graph of y = 3 arctan(x) Graph of y = 3arcsec(x) Graph of y = 3 + arcsin(x) Graph of y = 3 + arccos(x) Graph of y = 3 + arctan(x) Graph of y = 3 + arcsec(x) Graph of y = 3 + arcsin(x)

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17 Graph of y = 3 + arccos(x) Graph of y = 3 + arctan(x) Graph of y = 3 + arcsec(x) Compute lim arcsin(x) x 1 + Compute lim arcsin(x) x 1 Compute lim x 0 arcsin(x) Compute lim arccos(x) x 1 + Compute lim arccos(x) x 1 Compute lim x 0 arccos(x) Compute lim arctan(x) x

18 Draw the graph of y = arcsin(x). Draw the graph of y = arcsin(x). Draw the graph of y = arcsin(x). Draw the graph of y = arcsin(x). Draw the graph of y = arcsin(x). Draw the graph of y = arctan(x). Draw the graph of y = arcsin(x).

19 Compute lim x arctan(x) Compute lim x 0 arctan(x) Math Flash cars Formula (5.2) From Calculus I Definition of Definite Integral of continuous function Math Flash cars Math Flash cars Flash Cars Changes Definition (Calculus I) Constant Rule (Calculus I) f (x) x c Constant Multiple Rule (Calculus I) Sum Rule (Calculus I) x cf(x) [f(x) + g(x)] x

20 Draw the graph of y = arcsin(x). Draw the graph of y = arctan(x). b a f(x) x = lim n n f(c k ) k i=1 I hope that these flashcars help. Mark Stankus has moife these flash cars to be specific to the course Math 142 at the California Polytechnic State University of San Luis Obispo. Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License. These cars are part of a project to create free, high quality, printable flash cars for mathematics stuy. You are free to share, moify, an istribute these for non-commercial use, as long as you mention the source URL. Suggestion for aitional material an corrections are welcome. Copyright c 2009 Djun. M. Kim. This work is license uner a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canaa License f(x + h) f(x) lim h 0 h This only works for constants c. f (x) + g (x) cf (x) This only works for constants c.

21 Difference Rule (Calculus I) Prouct Rule (Calculus I) [f(x) g(x)] x x [f(x)g(x)] Quotient Rule (Calculus I) Chain Rule (Calculus I) x [ ] f(x) g(x) x [f(g(x))] Derivative (Calculus I) Derivative (Calculus I) x [xn ] x sin x Derivative (Calculus I) Derivative (Calculus I) x cos x x sec x Derivative (Calculus I) Derivative (Calculus I) x csc x x tan x

22 f (x)g(x) + f(x)g (x) f (x) g (x) f (g(x))g (x) f (x)g(x) f(x)g (x) [g(x)] 2 cos x nx n 1 This only works for constants n. You learne this in Calculus I for rational numbers n. We will learn this in Calculus II for irrational numbers n, like π. sec x tan x sin x sec 2 x csc x cot x

23 Derivative (Calculus I) Derivative (7.3) x cot x x ex (base e) Derivative (7.3) Derivative (7.2) x ax (base a) x ln x Derivative (7.3) First Step (7.3) x log a x x (cos(x))x Definition (7.2) Formula (7.2) ln(x) x ln(x) Formula (7.2) Formula (7.2) x ln(u) x ln( x)

24 e x csc 2 x 1 x a x ln a This only works when c is a constant. Use cos(x) = e ln(x) laws of exponents. 1 x ln a This only works constants a. x ln(x) = 1 x x ln(x) = 1 1 t t x ln( x) = 1 x x ln(u) = 1 u u x

25 Formula (7.2) x ln( x ) Domain of ln(x) Domain of ln(u) Domain of ln( x) Domain of ln( x ) Domain of ln( u ) Graph of y = ln(x) Graph of y = ln( x) Graph of y = ln( x ) Compute lim ln(x) x

26 (0, ) x ln( x ) = 1 x (, 0) u (0, ) u 0 x 0 Does not make sense! Draw the graph of y = ln(x) to memorize this.

27 Compute lim x ln(x) Compute lim x 0 + ln(x) Compute lim x 0 ln(x) Compute lim ln( x) x Compute lim x ln( x) Compute lim x 0 + ln( x) Compute lim x 0 ln( x) Compute lim ln( x ) x Compute lim x ln( x ) Compute lim x 0 + ln( x )

28 Draw the graph of y = ln(x) to memorize this. Draw the graph of y = ln(x) to memorize this. Draw the graph of y = ln( x) to memorize this. Does not make sense! Draw the graph of y = ln(x) to memorize this. Does not make sense! Draw the graph of y = ln( x) to memorize this. Does not make sense! Draw the graph of y = ln( x) to memorize this. Draw the graph of y = ln( x ) to memorize this. Draw the graph of y = ln( x) to memorize this. Draw the graph of y = ln( x ) to memorize this. Draw the graph of y = ln( x ) to memorize this.

29 Definition (7.2) Compute lim x 0 ln( x ) e Definition (7.3) Formula (7.3) e x x ex Formula (7.3) Match It!(7.3) x eu Domain of e x Match It!(7.3) Match It!(7.3) Range of e x Graph of y = e x Match It!(7.3) Match It!(7.3) Graph of y = e 2x Graph of y = e x+4

30 The number e is efine such that ln(e) = 1. Draw the graph of y = ln( x ) to memorize this. x ex = e x The function which is inverse to y = ln(x). (, ) x eu = e u u x (0, )

31 Match It!(7.3) Match It!(7.3) Graph of y = e x 4 Graph of y = 2 + e x Match It!(7.3) Match It!(7.3) Graph of e x Compute lim x ex Match It!(7.3) Match It!(7.3) Compute lim e x x Compute lim x 0 + ex Match It!(7.3) Compute lim x 0 ex

32 0 Draw the graph of y = e x to memorize it. 1 Draw the graph of y = e x to memorize it. Draw the graph of y = e x to memorize it. 1 Draw the graph of y = e x to memorize it.