Drews Calculus BC Flashcards

Terms Definitions
Quotient Rule
slope of horizontal line
Product Rule
uv' + vu'
absolute value of velocity
slope of vertical line
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))
When f '(x) is negative, f(x) is
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-&gt; a of f(x) exists
3. lim as x-&gt;a of f(x)=f(a)
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