# Drews Calculus BC Flashcards

Terms Definitions
 Integral Accumulation (d/dx)(tan(x)) (sec(x))^2 Quotient Rule (uv'-vu')/v² slope of horizontal line zero Product Rule uv' + vu' absolute value of velocity speed slope of vertical line undefined Convergent If lim(a(n) --> infinity) exists Integral of: (sec(x))^2 dx tan(x) + c Trapezoidal Rule Area of Trapezoid= ΔX/2 (f(Xk)+f(Xk+1)) Tn=ΔX/2(f(Xo)+2f(X1)+2f(X2)+...+2f(Xn-1)+f(Xn)) When f '(x) is negative, f(x) is decreasing y = ln(x), y' = y' = 1/x y = sec(x), y' = y' = sec(x)tan(x) 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) left riemann sum use rectangles with left-endpoints to evaluate integral (estimate area) Particle is moving to the left/down velocity is negative y = a^x, y' = y' = a^x ln(a) derivative of parametrically defined curve x(t) and y(t) dy/dx = dy/dt / dx/dt When f '(x) changes fro positive to negative, f(x) has a relative maximum y = cot⁻¹(x), y' = y' = -1/(1 + x²) First Derivative (Parametrics) dy/dx = (dy/dt)/(dx/dt) where dx/dt != 0 definite integral has limits a & b, find antiderivative, F(b) - F(a) y = sin⁻¹(x), y' = y' = 1/√(1 - x²) geometric series test general term = a₁r^n, converges if -1 < r < 1 Derivatives of Trig functions (sinx)' = cosx (cosx)' = -sinx (tanx)' = sec^2x (cotx)' = -csc^2x (secx)' = secxtanx (cscx)' = -cscxcotx When is a function not differentiable corner, cusp, vertical tangent, discontinuity To find absolute maximum on closed interval [a, b], you must consider... critical points and endpoints y = ln(x)/x², state rule used to find derivative quotient rule Alternate definition of derivative limit as x approaches a of [f(x)-f(a)]/(x-a) y = x cos(x), state rule used to find derivative product rule Theorem: If lim of |a(n)| = 0... lim of a(n) = 0 y = log (base a) x, y' = y' = 1/(x lna) Extreme Value Theorem If f is a continuous function on a closed interval [a,b], then f has both a maximum and minimum value on the interval 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 If f '(x) = 0 and f"(x) > 0, f(x) has a relative minimum A function is continuous if... 1. f(a) exists 2. lim as x-> a of f(x) exists 3. lim as x->a of f(x)=f(a)
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