# AP Calculus BC Exam_1 Flashcards

Terms Definitions
 rate derivative (d/dx)(sec(x)) sec(x)*tan(x) Quotient Rule (uv'-vu')/v² area below x-axis is negative y = cos²(3x) chain rule slope of vertical line undefined slope of horizontal line zero Product Rule uv' + vu' [(h1 - h2)/2]*base area of trapezoid Midpoint Rule f(a+ΔX(k+1/2)) = f((Xk+Xk+1)/2) Total Area: Mn=ΔX(f((X0+X1)/2)+...f((Xn-1+Xn)/2) = ΔXsum(f((Xk+Xk+1)/2),n-1,k=0) methods of integration substitution, parts, partial fractions converges absolutely alternating series converges and general term converges with another test Integral of: (csc(x))^2 dx -cot(x) + c x = ? (Polar) x = r*cos(@) integral test if integral converges, series converges When f '(x) is positive, f(x) is increasing y = sec(x), y' = y' = sec(x)tan(x) y = tan(x), y' = y' = sec²(x) y = e^x, y' = y' = e^x Squeeze Theorem: Integral (1/x) = divergent, Integral (1/x^2) = convergent. If less than the former it is convergent, if more than the latter it is divergent. y = csc(x), y' = y' = -csc(x)cot(x) trapezoidal rule use trapezoids to evaluate integrals (estimate area) y = cos(x), y' = y' = -sin(x) arc length pi INT(a to b) SQRT(1 +(dy/dx)^2)dx Riemann Sums ΔXsum(f(Xk),n-1,k=0), where Xk is any sample ppoint in the kth subinterval right riemann sum use rectangles with right-endpoints to evaluate integrals (estimate area) Particle is moving to the right/up velocity is positive First Derivative Shows the rate of change over time Integral of: csc(x) dx ln|csc(x) - cot(x)| + c left riemann sum use rectangles with left-endpoints to evaluate integral (estimate area) When f '(x) is increasing, f(x) is concave up 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) To draw a slope field, plug (x,y) coordinates into differential equation, draw short segments representing slope at each point y = tan⁻¹(x), y' = y' = 1/(1 + x²) 6th degree Taylor Polynomial polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative When f '(x) changes from negative to positive, f(x) has a relative minimum y = sin⁻¹(x), y' = y' = 1/√(1 - x²) l'Hospital's Rule lim (f(x)) = lim (f'(x)) = lim (f''(x)) ... indeterminate forms 0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰ y = cot⁻¹(x), y' = y' = -1/(1 + x²) Which functions are continuous everywhere? Polynomial, Rational, functions made up of continuous functions i.e. f + g , f - g , f * g, f/g ( if g dne 0) Implicit Differentiation 1. Separate y and x terms 2. Differentiate each term of the equation with respect to x 3. Factor out (dy/dx) on the left side of the equation 4. Solve for (dy/dx) dP/dt = kP(M - P) logistic differential equation, M = carrying capacity Formal definition of derivative limit as h approaches 0 of [f(a+h)-f(a)]/h Average Rate of Change Slope of secant line between two points, use to estimate instantanous rate of change at a point. alternating series test lim as n approaches zero of general term = 0 and terms decrease, series converges length of parametric curve ∫ √ (dx/dt)² + (dy/dt)² over interval from a to b ratio test lim as n approaches ∞ of ratio of (n+1) term/nth term > 1, series converges Instantenous Rate of Change Slope of tangent line at a point, value of derivative at a point use integration by parts when two different types of functions are multiplied Theorem: If the series sigma a(n) from 1 - infinite is convergent... lim a(n) = 0 given velocity vectors dx/dt and dy/dt, find speed √(dx/dt)² + (dy/dt)² not an integral! Derivatives of Exponential and Logarithmic Functions Let u be a differentiable function. (e^u)' = e^u (a^u)' = a^u lnu (du/dx) use partial fractions to integrate when integrand is a rational function with a factorable denominator find interval of convergence use ratio test, set > 1 and solve absolute value equations, check endpoints 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 Sum & Difference Rules for derivatives You can add or subtract derivatives (g +/- f)' = g' +/- f' Second Derivative Test for Relative Extrema If f'(c)=0 and f''(c)<0, then f(c) is a relative maximum If f'(c)=0 and f''(c)> 0, then f(c) is a relative minimum If f'(c)=0 and f''(c)=0, then f(c) is a relative maximum average value of f(x) = 1/(b-a) ∫ f(x) dx on interval a to b area inside polar curve 1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta For geometric series: sigma (a*r^(n-1)) from 1 to infinite... Convergent if |r| < 1 and sum is a/(1-r) Divergent if |r| >= 1 If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) = g'(x) = f(x) Test for Increasing/Decreasing Functions If f'(x) < 0 on [a,b] , then f(x) is decreasing on [a,b] If f'(x) > 0 on [a,b] , then f(x) is increasing on [a,b] If f'(x) = 0 on [a,b] , then f(x) is a constant on [a,b] Fundamental Theorem of Calculus ∫ f(x) dx on interval a to b = F(b) - F(a) First Derivative Test for Relative Extrema If f'(x) changes from pos to neg at x=c, then f has a relative maximum at c If f'(x) changes from neg to pos at x=c, then f has a relative minimum at c 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 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
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