Statesville Christian School AP Calculus Class
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| 161381118 | 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. | |
| 161381119 | Average Rate of Change | Slope of secant line between two points, use to estimate instantanous rate of change at a point. | |
| 161381120 | Instantenous Rate of Change | Slope of tangent line at a point, value of derivative at a point | |
| 161393670 | Formal definition of derivative | limit as h approaches 0 of [f(a+h)-f(a)]/h | |
| 161393671 | Alternate definition of derivative | limit as x approaches a of [f(x)-f(a)]/(x-a) | |
| 161399813 | When f '(x) is positive, f(x) is | increasing | |
| 161399814 | When f '(x) is negative, f(x) is | decreasing | |
| 161399815 | When f '(x) changes from negative to positive, f(x) has a | relative minimum | |
| 161399816 | When f '(x) changes fro positive to negative, f(x) has a | relative maximum | |
| 161399817 | When f '(x) is increasing, f(x) is | concave up | |
| 161399818 | When f '(x) is decreasing, f(x) is | concave down | |
| 161399819 | When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a | point of inflection | |
| 161399820 | When is a function not differentiable | corner, cusp, vertical tangent, discontinuity | |
| 161399821 | Product Rule | uv' + vu' | |
| 161399822 | Quotient Rule | (uv'-vu')/v² | |
| 161399823 | Chain Rule | f '(g(x)) g'(x) | |
| 161399824 | y = x cos(x), state rule used to find derivative | product rule | |
| 161399825 | y = ln(x)/x², state rule used to find derivative | quotient rule | |
| 161399826 | y = cos²(3x) | chain rule | |
| 161399827 | Particle is moving to the right/up | velocity is positive | |
| 161399828 | Particle is moving to the left/down | velocity is negative | |
| 161399829 | absolute value of velocity | speed | |
| 161399830 | y = sin(x), y' = | y' = cos(x) | |
| 161399831 | y = cos(x), y' = | y' = -sin(x) | |
| 161399832 | y = tan(x), y' = | y' = sec²(x) | |
| 161420341 | y = csc(x), y' = | y' = -csc(x)cot(x) | |
| 161420342 | y = sec(x), y' = | y' = sec(x)tan(x) | |
| 161420343 | y = cot(x), y' = | y' = -csc²(x) | |
| 161420344 | y = sin⁻¹(x), y' = | y' = 1/√(1 - x²) | |
| 161420345 | y = cos⁻¹(x), y' = | y' = -1/√(1 - x²) | |
| 161420346 | y = tan⁻¹(x), y' = | y' = 1/(1 + x²) | |
| 161420347 | y = cot⁻¹(x), y' = | y' = -1/(1 + x²) | |
| 161420348 | y = e^x, y' = | y' = e^x | |
| 161420349 | y = a^x, y' = | y' = a^x ln(a) | |
| 161420350 | y = ln(x), y' = | y' = 1/x | |
| 161420351 | y = log (base a) x, y' = | y' = 1/(x lna) | |
| 161420352 | To find absolute maximum on closed interval [a, b], you must consider... | critical points and endpoints | |
| 161420353 | 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) | |
| 161420354 | If f '(x) = 0 and f"(x) > 0, | f(x) has a relative minimum | |
| 161420355 | If f '(x) = 0 and f"(x) < 0, | f(x) has a relative maximum | |
| 161420356 | Linearization | use tangent line to approximate values of the function | |
| 161420357 | rate | derivative | |
| 161420358 | left riemann sum | use rectangles with left-endpoints to evaluate integral (estimate area) | |
| 161420359 | right riemann sum | use rectangles with right-endpoints to evaluate integrals (estimate area) | |
| 161420360 | trapezoidal rule | use trapezoids to evaluate integrals (estimate area) | |
| 161420361 | [(h1 - h2)/2]*base | area of trapezoid | |
| 161420362 | definite integral | has limits a & b, find antiderivative, F(b) - F(a) | |
| 161420363 | indefinite integral | no limits, find antiderivative + C, use inital value to find C | |
| 161420364 | area under a curve | ∫ f(x) dx integrate over interval a to b | |
| 161427773 | area above x-axis is | positive | |
| 161427774 | area below x-axis is | negative | |
| 161427775 | average value of f(x) | = 1/(b-a) ∫ f(x) dx on interval a to b | |
| 161427776 | If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = | g'(x) = f(x) | |
| 161427777 | Fundamental Theorem of Calculus | ∫ f(x) dx on interval a to b = F(b) - F(a) | |
| 161427778 | To find particular solution to differential equation, dy/dx = x/y | separate variables, integrate + C, use initial condition to find C, solve for y | |
| 161427779 | To draw a slope field, | plug (x,y) coordinates into differential equation, draw short segments representing slope at each point | |
| 161427780 | slope of horizontal line | zero | |
| 161427781 | slope of vertical line | undefined | |
| 161427782 | methods of integration | substitution, parts, partial fractions | |
| 161427783 | use substitution to integrate when | a function and it's derivative are in the integrand | |
| 161427784 | use integration by parts when | two different types of functions are multiplied | |
| 161427785 | ∫ u dv = | uv - ∫ v du | |
| 161427786 | use partial fractions to integrate when | integrand is a rational function with a factorable denominator | |
| 161427787 | dP/dt = kP(M - P) | logistic differential equation, M = carrying capacity | |
| 161427788 | P = M / (1 + Ae^(-Mkt)) | logistic growth equation | |
| 161427789 | 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 | |
| 161440797 | 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 | |
| 161440798 | given v(t) find displacement | ∫ v(t) over interval a to b | |
| 161440799 | given v(t) find total distance travelled | ∫ abs[v(t)] over interval a to b | |
| 161440800 | 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 | |
| 161440801 | 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 | |
| 161440802 | volume of solid of revolution - no washer | π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution | |
| 161440803 | 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 | |
| 161440804 | length of curve | ∫ √(1 + (dy/dx)²) dx over interval a to b | |
| 161440805 | L'Hopitals rule | use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit | |
| 161440806 | indeterminate forms | 0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰ | |
| 161440807 | 6th degree Taylor Polynomial | polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative | |
| 161440808 | Taylor series | polynomial with infinite number of terms, includes general term | |
| 161440809 | nth term test | if terms grow without bound, series diverges | |
| 161440810 | alternating series test | lim as n approaches zero of general term = 0 and terms decrease, series converges | |
| 161440811 | converges absolutely | alternating series converges and general term converges with another test | |
| 161440812 | converges conditionally | alternating series converges and general term diverges with another test | |
| 161440813 | ratio test | lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges | |
| 161440814 | find interval of convergence | use ratio test, set > 1 and solve absolute value equations, check endpoints | |
| 161440815 | find radius of convergence | use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint | |
| 161440816 | integral test | if integral converges, series converges | |
| 161440817 | 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 | |
| 161440818 | geometric series test | general term = a₁r^n, converges if -1 < r < 1 | |
| 161440819 | p-series test | general term = 1/n^p, converges if p > 1 | |
| 161440820 | derivative of parametrically defined curve x(t) and y(t) | dy/dx = dy/dt / dx/dt | |
| 161440821 | 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 | |
| 161440822 | length of parametric curve | ∫ √ (dx/dt)² + (dy/dt)² over interval from a to b | |
| 161440823 | given velocity vectors dx/dt and dy/dt, find speed | √(dx/dt)² + (dy/dt)² not an integral! | |
| 161440824 | given velocity vectors dx/dt and dy/dt, find total distance travelled | ∫ √ (dx/dt)² + (dy/dt)² over interval from a to b | |
| 161440825 | area inside polar curve | 1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta | |
| 161440826 | 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. |
