AP Calculus BC Flashcards
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| 1130394051 | Intermediate Value Theorem | If f(1)=-4 and f(6)=9, then there must be a x-value between 1 and 6 where f crosses the x-axis. | 1 | |
| 1130394052 | Average Rate of Change | Slope of secant line between two points, use to estimate instantanous rate of change at a point. | 2 | |
| 1130394053 | Instantenous Rate of Change | Slope of tangent line at a point, value of derivative at a point | 3 | |
| 1130394054 | Formal definition of derivative | limit as h approaches 0 of [f(a+h)-f(a)]/h | 4 | |
| 1130394055 | Alternate definition of derivative | limit as x approaches a of [f(x)-f(a)]/(x-a) | 5 | |
| 1130394056 | When f '(x) is positive, f(x) is | increasing | 6 | |
| 1130394057 | When f '(x) is negative, f(x) is | decreasing | 7 | |
| 1130394058 | When f '(x) changes from negative to positive, f(x) has a | relative minimum | 8 | |
| 1130394059 | When f '(x) changes fro positive to negative, f(x) has a | relative maximum | 9 | |
| 1130394060 | When f '(x) is increasing, f(x) is | concave up | 10 | |
| 1130394061 | When f '(x) is decreasing, f(x) is | concave down | 11 | |
| 1130394062 | When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a | point of inflection | 12 | |
| 1130394063 | When is a function not differentiable | corner, cusp, vertical tangent, discontinuity | 13 | |
| 1130394064 | Product Rule | uv' + vu' | 14 | |
| 1130394065 | Quotient Rule | (uv'-vu')/v² | 15 | |
| 1130394066 | Chain Rule | f '(g(x)) g'(x) | 16 | |
| 1130394067 | y = x cos(x), state rule used to find derivative | product rule | 17 | |
| 1130394068 | y = ln(x)/x², state rule used to find derivative | quotient rule | 18 | |
| 1130394069 | y = cos²(3x) | chain rule | 19 | |
| 1130394070 | Particle is moving to the right/up | velocity is positive | 20 | |
| 1130394071 | Particle is moving to the left/down | velocity is negative | 21 | |
| 1130394072 | absolute value of velocity | speed | 22 | |
| 1130394073 | y = sin(x), y' = | y' = cos(x) | 23 | |
| 1130394074 | y = cos(x), y' = | y' = -sin(x) | 24 | |
| 1130394075 | y = tan(x), y' = | y' = sec²(x) | 25 | |
| 1130394076 | y = csc(x), y' = | y' = -csc(x)cot(x) | 26 | |
| 1130394077 | y = sec(x), y' = | y' = sec(x)tan(x) | 27 | |
| 1130394078 | y = cot(x), y' = | y' = -csc²(x) | 28 | |
| 1130394079 | y = sin⁻¹(x), y' = | y' = 1/√(1 - x²) | 29 | |
| 1130394080 | y = cos⁻¹(x), y' = | y' = -1/√(1 - x²) | 30 | |
| 1130394081 | y = tan⁻¹(x), y' = | y' = 1/(1 + x²) | 31 | |
| 1130394082 | y = cot⁻¹(x), y' = | y' = -1/(1 + x²) | 32 | |
| 1130394083 | y = e^x, y' = | y' = e^x | 33 | |
| 1130394084 | y = a^x, y' = | y' = a^x ln(a) | 34 | |
| 1130394085 | y = ln(x), y' = | y' = 1/x | 35 | |
| 1130394086 | y = log (base a) x, y' = | y' = 1/(x lna) | 36 | |
| 1130394087 | To find absolute maximum on closed interval [a, b], you must consider... | critical points and endpoints | 37 | |
| 1130394088 | mean value theorem | if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a) | 38 | |
| 1130394089 | If f '(x) = 0 and f"(x) > 0, | f(x) has a relative minimum | 39 | |
| 1130394090 | If f '(x) = 0 and f"(x) < 0, | f(x) has a relative maximum | 40 | |
| 1130394091 | Linearization | use tangent line to approximate values of the function | 41 | |
| 1130394092 | rate | derivative | 42 | |
| 1130394093 | left riemann sum | use rectangles with left-endpoints to evaluate integral (estimate area) | 43 | |
| 1130394094 | right riemann sum | use rectangles with right-endpoints to evaluate integrals (estimate area) | 44 | |
| 1130394095 | trapezoidal rule | use trapezoids to evaluate integrals (estimate area) | 45 | |
| 1130394096 | [(h1 - h2)/2]*base | area of trapezoid | 46 | |
| 1130394097 | definite integral | has limits a & b, find antiderivative, F(b) - F(a) | 47 | |
| 1130394098 | indefinite integral | no limits, find antiderivative + C, use inital value to find C | 48 | |
| 1130394099 | area under a curve | ∫ f(x) dx integrate over interval a to b | 49 | |
| 1130394100 | area above x-axis is | positive | 50 | |
| 1130394101 | area below x-axis is | negative | 51 | |
| 1130394102 | average value of f(x) | = 1/(b-a) ∫ f(x) dx on interval a to b | 52 | |
| 1130394103 | If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = | g'(x) = f(x) | 53 | |
| 1130394104 | Fundamental Theorem of Calculus | ∫ f(x) dx on interval a to b = F(b) - F(a) | 54 | |
| 1130394105 | To find particular solution to differential equation, dy/dx = x/y | separate variables, integrate + C, use initial condition to find C, solve for y | 55 | |
| 1130394106 | To draw a slope field, | plug (x,y) coordinates into differential equation, draw short segments representing slope at each point | 56 | |
| 1130394107 | slope of horizontal line | zero | 57 | |
| 1130394108 | slope of vertical line | undefined | 58 | |
| 1130394109 | methods of integration | substitution, parts, partial fractions | 59 | |
| 1130394110 | use substitution to integrate when | a function and it's derivative are in the integrand | 60 | |
| 1130394111 | use integration by parts when | two different types of functions are multiplied | 61 | |
| 1130394112 | ∫ u dv = | uv - ∫ v du | 62 | |
| 1130394113 | use partial fractions to integrate when | integrand is a rational function with a factorable denominator | 63 | |
| 1130394114 | dP/dt = kP(M - P) | logistic differential equation, M = carrying capacity | 64 | |
| 1130394115 | P = M / (1 + Ae^(-Mkt)) | logistic growth equation | 65 | |
| 1130394116 | given rate equation, R(t) and inital condition when t = a, R(t) = y₁ find final value when t = b | y₁ + Δy = y Δy = ∫ R(t) over interval a to b | 66 | |
| 1130394117 | given v(t) and initial position t = a, find final position when t = b | s₁+ Δs = s Δs = ∫ v(t) over interval a to b | 67 | |
| 1130394118 | given v(t) find displacement | ∫ v(t) over interval a to b | 68 | |
| 1130394119 | given v(t) find total distance travelled | ∫ abs[v(t)] over interval a to b | 69 | |
| 1130394120 | area between two curves | ∫ f(x) - g(x) over interval a to b, where f(x) is top function and g(x) is bottom function | 70 | |
| 1130394121 | volume of solid with base in the plane and given cross-section | ∫ A(x) dx over interval a to b, where A(x) is the area of the given cross-section in terms of x | 71 | |
| 1130394122 | volume of solid of revolution - no washer | π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution | 72 | |
| 1130394123 | volume of solid of revolution - washer | π ∫ R² - r² dx over interval a to b, where R = distance from outside curve to axis of revolution, r = distance from inside curve to axis of revolution | 73 | |
| 1130394124 | length of curve | ∫ √(1 + (dy/dx)²) dx over interval a to b | 74 | |
| 1130394125 | L'Hopitals rule | use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit | 75 | |
| 1130394126 | indeterminate forms | 0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰ | 76 | |
| 1130394127 | 6th degree Taylor Polynomial | polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative | 77 | |
| 1130394128 | Taylor series | polynomial with infinite number of terms, includes general term | 78 | |
| 1130394129 | nth term test | if terms grow without bound, series diverges | 79 | |
| 1130394130 | alternating series test | lim as n approaches zero of general term = 0 and terms decrease, series converges | 80 | |
| 1130394131 | converges absolutely | alternating series converges and general term converges with another test | 81 | |
| 1130394132 | converges conditionally | alternating series converges and general term diverges with another test | 82 | |
| 1130394133 | ratio test | lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges | 83 | |
| 1130394134 | find interval of convergence | use ratio test, set > 1 and solve absolute value equations, check endpoints | 84 | |
| 1130394135 | find radius of convergence | use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint | 85 | |
| 1130394136 | integral test | if integral converges, series converges | 86 | |
| 1130394137 | limit comparison test | if lim as n approaches ∞ of ratio of comparison series/general term is positive and finite, then series behaves like comparison series | 87 | |
| 1130394138 | geometric series test | general term = a₁r^n, converges if -1 < r < 1 | 88 | |
| 1130394139 | p-series test | general term = 1/n^p, converges if p > 1 | 89 | |
| 1130394140 | derivative of parametrically defined curve x(t) and y(t) | dy/dx = dy/dt / dx/dt | 90 | |
| 1130394141 | second derivative of parametrically defined curve | find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt | 91 | |
| 1130394142 | length of parametric curve | ∫ √ (dx/dt)² + (dy/dt)² over interval from a to b | 92 | |
| 1130394143 | given velocity vectors dx/dt and dy/dt, find speed | √(dx/dt)² + (dy/dt)² not an integral! | 93 | |
| 1130394144 | given velocity vectors dx/dt and dy/dt, find total distance travelled | ∫ √ (dx/dt)² + (dy/dt)² over interval from a to b | 94 | |
| 1130394145 | area inside polar curve | 1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta | 95 | |
| 1130394146 | area inside one polar curve and outside another polar curve | 1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta. | 96 |