Calculus AB Exam Flashcards
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| 382023814 | 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. | |
| 382023815 | Average Rate of Change | Slope of secant line between two points, use to estimate instantanous rate of change at a point. | |
| 382023816 | Instantenous Rate of Change | Slope of tangent line at a point, value of derivative at a point | |
| 382023817 | Formal definition of derivative | limit as h approaches 0 of [f(a+h)-f(a)]/h | |
| 382023818 | Alternate definition of derivative | limit as x approaches a of [f(x)-f(a)]/(x-a) | |
| 382023819 | When f '(x) is positive, f(x) is | increasing | |
| 382023820 | When f '(x) is negative, f(x) is | decreasing | |
| 382023821 | When f '(x) changes from negative to positive, f(x) has a | relative minimum | |
| 382023822 | When f '(x) changes fro positive to negative, f(x) has a | relative maximum | |
| 382023823 | When f '(x) is increasing, f(x) is | concave up | |
| 382023824 | When f '(x) is decreasing, f(x) is | concave down | |
| 382023825 | When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a | point of inflection | |
| 382023826 | When is a function not differentiable | corner, cusp, vertical tangent, discontinuity | |
| 382023827 | Product Rule | uv' + vu' | |
| 382023828 | Quotient Rule | (uv'-vu')/v² | |
| 382023829 | Chain Rule | f '(g(x)) g'(x) | |
| 382023830 | y = x cos(x), state rule used to find derivative | product rule | |
| 382023831 | y = ln(x)/x², state rule used to find derivative | quotient rule | |
| 382023832 | y = cos²(3x) | chain rule | |
| 382023833 | Particle is moving to the right/up | velocity is positive | |
| 382023834 | Particle is moving to the left/down | velocity is negative | |
| 382023835 | absolute value of velocity | speed | |
| 382023836 | y = sin(x), y' = | y' = cos(x) | |
| 382023837 | y = cos(x), y' = | y' = -sin(x) | |
| 382023838 | y = tan(x), y' = | y' = sec²(x) | |
| 382023839 | y = csc(x), y' = | y' = -csc(x)cot(x) | |
| 382023840 | y = sec(x), y' = | y' = sec(x)tan(x) | |
| 382023841 | y = cot(x), y' = | y' = -csc²(x) | |
| 382023842 | y = sin⁻¹(x), y' = | y' = 1/√(1 - x²) | |
| 382023843 | y = cos⁻¹(x), y' = | y' = -1/√(1 - x²) | |
| 382023844 | y = tan⁻¹(x), y' = | y' = 1/(1 + x²) | |
| 382023845 | y = cot⁻¹(x), y' = | y' = -1/(1 + x²) | |
| 382023846 | y = e^x, y' = | y' = e^x | |
| 382023847 | y = a^x, y' = | y' = a^x ln(a) | |
| 382023848 | y = ln(x), y' = | y' = 1/x | |
| 382023849 | y = log (base a) x, y' = | y' = 1/(x lna) | |
| 382023850 | To find absolute maximum on closed interval [a, b], you must consider... | critical points and endpoints | |
| 382023851 | 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) | |
| 382023852 | If f '(x) = 0 and f"(x) > 0, | f(x) has a relative minimum | |
| 382023853 | If f '(x) = 0 and f"(x) < 0, | f(x) has a relative maximum | |
| 382023854 | Linearization | use tangent line to approximate values of the function | |
| 382023855 | rate | derivative | |
| 382023856 | left riemann sum | use rectangles with left-endpoints to evaluate integral (estimate area) | |
| 382023857 | right riemann sum | use rectangles with right-endpoints to evaluate integrals (estimate area) | |
| 382023858 | trapezoidal rule | use trapezoids to evaluate integrals (estimate area) | |
| 382023859 | [(h1 - h2)/2]*base | area of trapezoid | |
| 382023860 | definite integral | has limits a & b, find antiderivative, F(b) - F(a) | |
| 382023861 | indefinite integral | no limits, find antiderivative + C, use inital value to find C | |
| 382023862 | area under a curve | ∫ f(x) dx integrate over interval a to b | |
| 382023863 | area above x-axis is | positive | |
| 382023864 | area below x-axis is | negative | |
| 382023865 | average value of f(x) | = 1/(b-a) ∫ f(x) dx on interval a to b | |
| 382023866 | If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = | g'(x) = f(x) | |
| 382023867 | Fundamental Theorem of Calculus | ∫ f(x) dx on interval a to b = F(b) - F(a) | |
| 382023868 | To find particular solution to differential equation, dy/dx = x/y | separate variables, integrate + C, use initial condition to find C, solve for y | |
| 382023869 | To draw a slope field, | plug (x,y) coordinates into differential equation, draw short segments representing slope at each point | |
| 382023870 | slope of horizontal line | zero | |
| 382023871 | slope of vertical line | undefined | |
| 382023872 | methods of integration | substitution, parts, partial fractions | |
| 382023873 | use substitution to integrate when | a function and it's derivative are in the integrand | |
| 382023874 | use integration by parts when | two different types of functions are multiplied | |
| 382023875 | ∫ u dv = | uv - ∫ v du | |
| 382023876 | use partial fractions to integrate when | integrand is a rational function with a factorable denominator | |
| 382023877 | dP/dt = kP(M - P) | logistic differential equation, M = carrying capacity | |
| 382023878 | P = M / (1 + Ae^(-Mkt)) | logistic growth equation | |
| 382023879 | 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 | |
| 382023880 | 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 | |
| 382023881 | given v(t) find displacement | ∫ v(t) over interval a to b | |
| 382023882 | given v(t) find total distance travelled | ∫ abs[v(t)] over interval a to b | |
| 382023883 | 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 | |
| 382023884 | 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 | |
| 382023885 | volume of solid of revolution - no washer | π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution | |
| 382023886 | 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 | |
| 382023887 | length of curve | ∫ √(1 + (dy/dx)²) dx over interval a to b | |
| 382023888 | L'Hopitals rule | use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit | |
| 382023889 | sin^2 x + cos^2 x | 1 | |
| 382023890 | 1 + tan^2 x | sec^2 x | |
| 382023891 | 1 + cot^2 x | csc^2 x | |
| 382023892 | sin(-x) | -sin x | |
| 382023893 | cos(-x) | cos x | |
| 382023894 | tan(-x) | -tan x | |
| 382023895 | sin(A + B) | sin A cos B + sin B cos A | |
| 382023896 | sin(A - B) | sin A cos B - sin B cos A | |
| 382023897 | cos(A + B) | cos A cos B - sin A sin B | |
| 382023898 | cos(A - B) | cos A cos B + sin A sin B | |
| 382023899 | sin 2x | 2 sin x cos x | |
| 382023900 | cos 2x (1) | cos^2 x - sin^2 x | |
| 382023901 | cos 2x (2) | 2cos^2 x - 1 | |
| 382023902 | cos 2x (3) | 1 - 2sin^2 x | |
| 382023903 | tan x | 1/ cot x | |
| 382023904 | cot x | 1/ tan x | |
| 382023905 | sec x | 1/ cos x | |
| 382023906 | csc x | 1/ sin x | |
| 382023907 | cos(pi/2 - x) | sin x | |
| 382023908 | sin(pi/2 - x) | cos x | |
| 382023909 | d/dx (x^n) | nx^n - 1 | |
| 382023910 | d/dx (fg) | fg' + gf' | |
| 382023911 | d/dx (f/g) | (gf' - fg')/g^2 | |
| 382023912 | d/dx f(g(x)) | f'(g(x))g'(x) | |
| 382023913 | d/dx (sin x) | cos x | |
| 382023914 | d/dx (cos x) | - sin x | |
| 382023915 | d/dx (tan x) | sec^2 x | |
| 382023916 | d/dx (cot x) | - csc^2 x | |
| 382023917 | d/dx (sec x) | sec x tan x | |
| 382023918 | d/dx (csc x) | -csc x cot x | |
| 382023919 | d/dx (e^x) | e^x | |
| 382023920 | d/dx (a^x) | a^x ln a | |
| 382023921 | d/dx (ln x) | 1/x | |
| 382023922 | d/dx (Arcsin x) | 1/(sqrt(1 - x^2)) | |
| 382023923 | d/dx (Arctan x) | 1/(1 + x^2) | |
| 382023924 | ∫ a dx | ax + c | |
| 382023925 | ∫ 1/x dx | ln | x | + c | |
| 382023926 | ∫ e^x ds | e^x + c | |
| 382023927 | ∫ a^x dx | a^x/ ln a + c | |
| 382023928 | ∫ ln x dx | x ln x - x + c | |
| 382023929 | ∫ sin x dx | -cos x + c | |
| 382023930 | ∫ cos x dx | sin x + c | |
| 382023931 | ∫ tan x dx (1) | ln |sec x| + c | |
| 382023932 | ∫ tan x dx (2) | -ln |cos x| + c | |
| 382023933 | ∫ cot x dx | ln |sin x| + c | |
| 382023934 | ∫ sec x dx | ln |sec x + tan x| + c | |
| 382023935 | ∫ csc x dx | -ln |csc x + cot x| + c | |
| 382023936 | ∫ sec^2 x dx | tan x + c | |
| 382023937 | ∫ sec x tan x dx | sec x + c | |
| 382023938 | ∫ csc^2 x dx | - cot x + c | |
| 382023939 | ∫ csc x cot x dx | - csc x + c | |
| 382023940 | ∫ tan^2 x dx | tan x - x + c | |
| 382023941 | ∫ dx/(a^2 + x^2) | 1/a Arctan (x/a) + c | |
| 382023942 | ∫ dx/(sqrt(a^2 - x^2)) | Arcsin (x/a) + c |