Flash Card Construction Instructions

Transcription

1 Flash Car Construction Instructions *** THESE CARDS ARE FOR CALCULUS HONORS, AP CALCULUS AB AND AP CALCULUS BC. AP CALCULUS BC WILL HAVE ADDITIONAL CARDS FOR THE COURSE (IN A SEPARATE FILE). The left column is the question an the right column is the answers. Cut out the flash cars an paste the question to one sie of a note car an the answer to the other sie. Be careful to paste the correct answer to its corresponing question

2 COMMON FORMULAS/TRIGONOMETRY/GEOMETRY Mipoint formula x 1 + x 2 2, y 1 + y 2 2 Distance formula (between 2 points) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Quaratic Formula b ± b 2 4ac 2a Pythagorean Theorem a 2 + b 2 = c 2 sinθ = opp hyp an y r an 1 cscθ cosθ = aj hyp an x r an 1 secθ tanθ = opp aj an y x an 1 cotθ

3 cotθ = aj opp an x y an 1 tanθ cscθ = hyp opp an r y an 1 sinθ secθ = hyp aj an r x an 1 cosθ Quotient Ientity tanu sinu cosu Quotient Ientity cotu cosu sinu Pythagorean Ientities sin 2 u + cos 2 u = tan 2 u = sec 2 u 1 + cot 2 u = csc 2 u Area of a Circle/ Circumference of a circle A = πr 2 C = 2πr Area of a Parallelogram A = bh

4 Area of a Trapezoi 1 2 h(b 1 + b 2 ) Area of a Triangle 1 2 bh triangle 1) Hypotenuse is 2 time short leg 2) Long leg is 3 times short leg triangle 1) Hypotenuse is 2 times leg 2) Two legs are equal sin 0 sin0 = 0 sin 30 sin 30 = 1 2 sin 45 sin 45 = 2 2 sin60 sin 60 = 3 2 sin90 sin90 = 1

5 cos0 cos0 = 1 cos 30 cos 30 = 3 2 cos 45 cos 45 = 2 2 cos60 cos60 = 1 2 cos90 cos90 = 0 tan0 tan 0 = 0 tan 30 tan 30 = 3 3 tan 45 tan 45 = 1 tan60 tan60 = 3 tan90 Unefine

6 sin(α + β) = sin(α + β) = sinα cosβ + cosα sin β sin(α β) = sin(α β) = sinα cosβ cosα sin β cos(α + β) = cos(α + β) = cosα cosβ sinα sinβ cos(α β) = cos(α β) = cosα cosβ + sinα sin β sin2θ = 2sinθ cosθ cos2θ = cos 2 θ sin 2 θ 2cos sin 2 θ Law of sines Law of cosines sin A a a sin A = = sin B b or b sin B = = sinc c c sinc a 2 = b 2 + c 2 2bccos A b 2 = a 2 + c 2 2accos B c 2 = a 2 + b 2 2abcos C Heron s Formula s(s a)(s b)(s c) s = a + b + c 2

7 What is a solution point. (x,y) pair that makes an equations with an x an y true P.1 How to fin x an y intercepts of an equation. x-intercept set y=0 an solve for x y-intercept set x=0 an solve for y P.1 What are the three types of symmetry? P.1 What are the 3 tests for symmetry? y-axis (replacing x with x yieling original equation) x-axis (replacing y with y yieling original equation) origin (replacing x with x an y with y yieling original equations y-axis x-axis origin P.1 How to fin the points of intersections of two equations? Simultaneously solving equations (elimination, substitution or using intersect feature of calculator P.1 The formula for fining the slope between two points? y 2 y 1 x 2 x 1 P.2 What are the 4 types of slope? positive, negative, zero, unefine P.2

8 What is the point slope form of the equation of a line? y y 1 = m(x x 1 ) P.2 What are the relationships of slopes between parallel lines an perpenicular lines. parallel lines (same slope), perpenicular lines (negative reciprocal slopes) P.2 How o you calculate an average rate of change? f (b) f (a) b a P.2 What is the slope-intercept equation of a line? y = mx + b P.2 What is the relationship between a relation an a function? Function has each x pointing to only one y value P.3 What oes one-to-one mean? each y value is pointe to by only one x- value P.3 What oes onto mean? range consists of all of Y P.3

9 How o you prove a graph is a function? passes the Vertical Line Test P.3 What are the 3 categories of elementary functions? P.3 P.3 What is the leaing coefficient test for polynomials? a. algebraic (polynomial, raical, rational) b. trigonometric c. exponential an logarithmic a. even exponent of leaing coefficient i. leaing coefficient > 0 up/up ii. leaing coefficient < 0 own/own b. o exponent of leaing coefficient iii. leaing coefficient > 0 own left/up right iv. leaing coefficient < 0 up left/own right What is an o function? (symmetric about origin) P.3 What is an even function? (y-axis symmetry) P.3 What is the relationship of the omain an range in inverse functions? The omains an ranges are swappe P.4 How can you etermine if a function has an inverse? Original function will pass the Horizontal Line Test P.4 How can you visually etermine of two functions are inverses of each other? The two functions will be reflecte about the line y = x P.4

10 P.4 What are the omains an ranges of arcsin? Domain: 1 x 1 Range y P.4 What are the omains an ranges of arccos? Domain: 1 x 1 Range 0 y π P.4 What are the omains an ranges of arctan? Domain: < x < Range < y < P.5 P.5 P.5 P.5 a a a (a ) (ab) 1 a a " a b P.5 a a a P.5 ( a b ) a b P.5 a 1 a

11 lne x P.5 e "# x P.5 P.5 What are the omains an ranges of lnx? Domain:(0, ) Range (, ) P.5 What are the omains an ranges of e? Domain:(, ) Range 0, )

12 What is the formula for fining a secant line? M sec = f (x + Δx) f (x) Δx What is the concept of a limit? If f(x) becomes arbitrarily close to a single number L as x approaches c from either sie the limit of f(x), as x approaches c, is L What is a generic efinition of a tangent line? A line that touches curve at one point 1.1 What are the 3 conitions that nee to be met for a limit to exist? 1.2 What are the 3 conitions where a limit fails to exist? 1.2 a. lim x a + b. lim x a c. lim x a + f (x)exists f (x)exists f (x) = lim x a f (x) a. unboune behavior (vertical asymptote) b. limit from the left not equal to the limit from the right c. oscillating behavior What is well-behave function? lim f (x) = f (c) x c 1.3 What are the 3 basic types of algebraic functions? a. polynomial b. rational c. raical 1.3 What are techniques for fining limits? 1.3 a. irect substitution (plug n chug) b. iviing out (factoring) c. rationalizing the numerator. make a table/graph

13 What are the ineterminate forms of a function? 0 0 or 1.3 sin x lim x 0 x cos x lim x 0 x lim (1 + x) e 1.3 What are the 3 conitions that nee to be met for continuity? 1.4 a. f(a) efine b. lim f (x) exists x a c. f(a) = lim f (x) x a What is the concept of a continuous function? when a graph can be rawn without lifting the pencil 1.4 What is the concept of everywhere continuous? continuous over the entire number line 1.4 What are 3 types of iscontinuity? a. hole b. infinite (vertical asymptote) c. jump 1.4 What is the concept of a one-sie limit? when only the limit from the left or the limit from the right of x=c is efine. 1.4

14 What are 5 types of functions that are continuous at every point in their omain? What oes the Intermeiate Value Theorem state? a. polynomial functions b. rational functions c. raical functions. trigonometric functions e. exponential an logarithmic If f is continuous on the close interval [a,b] an k is any number between f(a) an f(b), then there exists at least one number c in [a,b] such that f(c) =k 1.5 What is a vertical asymptote? Vertical line that is approache but never touche (en behavior) an is a result of the enominator of a rational expression being unefine How can you etermine the ifference between when a hole exists an a vertical asymptote exists? If you can cancel a factor out of enominator it is a hole What is a horizontal asymptote? lim c x lim c x Horizontal line that is approache but never touche (en behavior) an is a result of the enominator growing faster than the numerator lim e lim e 0

15 x > (sneaky technique) x < What are the 3 tests for etermining horizontal asymptotes? (sneaky technique) num exponent > en exponent, no asymptote num exponent < en exponent, y=0 num exponent = en exponent, y= leaingcoefficient leaingcoefficient

16 What is the efinition of the erivative of a function using limits? f '(x) = lim Δx 0 f (x + Δx) f (x) Δx 2.1 What is an alternate form of the erivative function using limits? f c = lim f x f(c) x c 2.1 What is the ifference quotient? f (x + Δx) f (x) Δx 2.1 What are the 3 cases where a erivative fails to exist? 2.1 Differentiation Rules: Constant Rule a. any point of iscontinuity b. cusp c. vertical tangent line x [c] = Differentiation Rules: Simple Power Rule x [xn ] = nx n Differentiation Rules: Constant Multiple Rule [cf (x)] = cf '(x) x 2.2 Differentiation Rules: Sum an Difference Rules [ f (x) ± g(x)] = f '(x) ± g'(x) x 2.2 [sin x] x cos x 2.2

17 [cos x] = x sin x 2.2 x [e ] e What is the stanar position function? [tan x] = x s t = 16t + V t + S -4.9 can be substitute if calculating in meters instea of feet sec 2 x [csc x] = x [sec x] = x csc x cot x sec x tan x 2.3 [cot x] = x Differentiation Rules: Prouct Rule csc 2 x f (x)g'(x)+ g(x) f '(x) first secon + secon first 2.3

18 2.3 Differentiation Rules: Quotient Rule g(x) f '(x) f (x)g'(x) g(x) 2 bottom top top bottom over bottom square 2.4 Differentiation Rules: Chain Rule f '(g(x))g'(x) outer inner (on t touch the stuff) Differentiation Rules: General Power Rule x [sinu] = nu n 1 u' (cosu)u' 2.4 x [cosu] = ( sinu)u' x [tanu] = x [cotu] = x [secu] = (sec 2 u)u' (csc 2 u)u' (secutanu)u' 2.4

19 x [cscu] = (cscucotu)u' 2.4 [ln x] x 1 x, x > [ln u ] x u u log x 1 lnx lnx or lna lna 2.4 x [a ] lna a 2.4 x [a ] lna a u x 2.4 x [log x] 1 lna x

20 x [log u] 1 u lna u x or u lna u 2.4 x [e ] e u 2.4 What is the explicit form of an equation? when an equation is solve for one variable 2.5 Inverse functions have what types of slopes at inverse pairs of points? reciprocal slopes 2.6 x [arcsinu] = u' 1 u x [arccosu] = u' 1 u x [arctanu] = u' 1+ u x [arccotu] = u' 1+ u x [arcsecu] = u u' u 2 1

21 x [arccscu] = u u' u What is a relate rate erivative usually taken with respect to? time 2.7 What is the formula for the volume of a cone? V = π 3 r h What is the formula for the volume of a sphere? What is a another name for a tangent line of approximation calle? V = 4 3 πr linear approximation 2.8 What metho uses a tangent line to approximate the y-values of a function? Newton s metho 2.8

22 What is a maximum? f(c) > all f(x) on an interval 3.1 What is a minimum? f(c) < all f(x) on an interval 3.1 What is the ifference between critical numbers an critical points? 3.1 What theorem state if f is continuous on a close interval [a,b], then f has both a minimum an a maximum on the interval 3.1 critical numbers are x-values an critical points are (x,y). Critical numbers are foun when f '(c) = 0 or where f '(c) oes not exist. Extreme Value Theorem Where oes the erivative fail to ientify possible extrema? enpoints 3.1 What oes Rolle s Theorem state? if f(a) = f(b) then there exists at least one number c in (a,b) such that f '(c) = What oes the Mean Value Theorem state? f '(c) = f (b) f (a) b a What are two major similarities between Rolle s Theorem an the Mean Value Theorem? Function must be 1) continuous an 2) ifferentiable 3.2 What is meant by increasing in terms of a erivative? f '(x) > 0 for all x in (a,b) 3.3

23 What is meant by ecreasing in terms of a erivative? f '(x) < 0 for all x in (a,b) 3.3 What is meant by constant in terms of a erivative? f '(x) = 0 for all x in (a,b) 3.3 What oes strictly monotonic mean? 3.3 What oes the first erivative test state? 3.3 How o you use the secon erivative to etermine concavity? 3.4 When a function is either increasing or ecreasing on entire interval a. if f '(x) changes from increasing to ecreasing at x =c then f '(c) is a relative maximum b. if f '(x) changes from ecreasing to increasing at x =c then f '(c) is a relative minimum c. if f '(x) oes not change signs at x =c then f '(c) is a neither a relative maximum or relative minimum a. if f ''(x) > 0, for all x in an interval f is concave upwar b. if f ''(x) < 0, for all x in an interval f is concave ownwar 3.4 What are points of inflection? where f ''(c) = 0 or f ''(c) is unefine (where a graph goes from concave upwar to concave ownwar or vice versa

24 How o you use the secon erivative to etermine relative extrema using critical numbers? 3.4 a. if f ''(c) > 0,then f(c) is a relative minimum b. if f ''(c) < 0,then f(c) is a relative maximum c. if f ''(c) = 0 then use must use the First Derivative Test 3.6 In optimization problems what is the equation that is to be optimize calle? primary equation What is a ifferential equation? What is the equation for a tangent line of approximation (linear approximation)? an equation that contains a erivative y = f c + f (c)(x c)

25 0x C 4.1 u = u + C kf (x)x k f (x)x 4.1 [ f (x) ± g(x)]x f (x)x ± g(x)x 4.1 x n x = x n+1 n +1 + C 4.1 cos x x = sin x + C 4.1 sin x x = cos x + C 4.1 (sec 2 x)x tan x + C 4.1 sec x tan x x sec x + C

26 (csc 2 x)x = cot x + C 4.1 csc x cot x x csc x + C 4.1 e x e + C 4.1 a x ( 1 lna )a + C x x ln x + C To change a general solution into a particular solution what is neee? an initial conition 4.2 a is what type of notation? sigma notation 4.2 c cn 4.2 i n(n + 1) 2

27 4.2 i n(n + 1)(2n + 1) i Left Rectangle Rule Right Rectangle Rule n (n + 1) 4 b a n (f(x ) + f x ) b a n (f(x ) + f x ) 4.3 The efinite integral as the area of a region f x x 4.3 f x x 0 f x x f x x 4.3 f x x with point c between a an b f x x + f x x 4.3

28 kf(x)x) k f x x 4.3 f x ± g x x f x x ± g x x Trapezoial Rule b a 2n [f x + 2f x + 2f(x ) + f(x ) ] 4.4 Funamental Theorem of Calculus f x x = F b F(a) 4.4 Mean Value Theorem For Integrals f x x = f(c)(b a) 4.4 Average value of a function 1 b a f x x 4.4 Secon Funamental Theorem of Calculus x [ f t t] = f x 4.4 Net Change Theorem F x = F b F(a)

29 4.5 u n u = u n+1 n +1 + C kf x x k f x x 4.5 f x x (even function) 2 f x x 4.5 f x x (o function) u u = ln u + C 4.6 a u u = 1 a u + C ln a 4.6 sinu u = cosu + C 4.6 cosu u = sinu + C 4.6 tanu u = ln cosu + C 4.6

30 cotu u = ln sinu + C 4.6 secu u = ln secu + tanu + C 4.6 cscu = ln cscu + cotu + C 4.6 sec 2 u u = tanu + C 4.6 csc 2 u u = cotu + C 4.6 secu tanu u = secu + C 4.6 cscu cotu u = cscu + C 4.6 u a 2 + u = 2 1 a arctan u a + C 4.7

31 u a 2 u 2 = arcsin u a + C 4.7 u u u 2 a = 2 1 a arcsec u a + C 4.7

32 What is a ifferential equation? an equation that inclues a erivative What is Euler s Metho? a numerical approach to approximating the particular solution to a ifferential equation What is the solution to a exponential growth or ecay problem? y = Ce " What is k in a half-life problem? ln ( 1 2 ) = k t What is the process of collecting all terms with x s an y s on opposite sies of the equal sign calle? separation of variables 5.2

33 How o you fin the area between two curves? 6.1 b a [ f (x) g(x)]x Disk Metho Horizontal Axis of Revolution Disk Metho Vertical Axis of Revolution Washer Metho Horizontal Axis of Revolution Washer Metho Vertical Axis of Revolution b b π [R(x)] 2 x a π [R(y)] 2 y c π ([R(x)] 2 [r(x)] 2 )x a π ([R(y)] 2 [r(y)] 2 )y c 6.2 Volume of soli with known cross section perpenicular to x-axis b a A(x)x 6.2 Volume of soli with known cross section perpenicular to y-axis c A(y)y