AP BC CALC
Lagrange Error Bound Def
Exact value = Approximate value + Remainder f(x) = P(x) + R(x)
Position, velocity, acceleration
F(t) = position, V(t) = F'(t) = velocity, A(t) = V'(t) = acceleration
A in P when limits aren't given
Find intersection points -> limits On calculator: 2nd trace, intersect
Nth Taylor Polynomial
If f(x) is a differentiable function, then an approximation of f centered about x = c can be modeled by Pn(x) = f(c) + f'(c)(x-c) + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ... + f^(n)(c)(x-c)^n/n! At the center P(x) = f(x)
Intermediate Value Theorem (IVT)
If f(x) is continuous on the closed interval [a,b], then f(x) will take on every value between f(a) and f(b)
IVT applications
If k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
Length of a Curve (Parametric)
L = ∫(a,b) sqrt((dx/dt)^2 + (dy/dt)^2)dt
Length of a Curve (Polar)
L = ∫(a,b) sqrt(r^2 + (dr/dtheta)^2)dtheta
Linearization
L(x) = f(a) + f'(a)(x-a)
Position of particle at x = b when you know P(0)
P(0) + ∫(0,b)V(x)dx initial position + displacement
Direct Comparison Test
Pick bn to compare to an Bn Converges: 0<=an<=bn (always positve and less than bn) Bn Diverges: 0<=bn<=an (always positive and greateer than bn)
Absolute value of V(x)
Speed
Speed inc/dec statement
Speed inc/dec on (interval) because velocity and acceleration signs match
Total distance traveled
Sum of the |xf-xi|s for each time interval. Each time interval is determined by the changes in sign of the velocity.
Trapezoidal Approximation Method
T = h/2(y0 + 2y1 + 2y2 + 2yn-1 + yn) n = number of subintervals h = subinterval length = (a-b)/n
Length of a Curve (Cartesian)
Terms of x: L = ∫(a,b) sqrt(1+(dy/dx)^2)dx Terms of y: L = ∫(c,d) sqrt(1+(dx/dy)^2)dy
Cone formula
V = 1/3πr^2h
Sphere formula
V = 4/3πr^3
Distance inc/dec statement
V>/<0 on (intervals) so the distance is inc/dec
when does speed increase and decrease
Velocity and acceleration have the same sign
Velocity inc/dec statement
Velocity inc/dec when a>/<0 on (interval)
Blob (Volumes of Revolution)
Volume = ∫(a, b) Area dx
Euler's Method
Yn=Yn-1+hF(Xn-1,Yn-1)dx the coordinate = the previous coordinate (1 step before) + the derivative of the previous coordinate*step size We start a step away from the starting point and use the starting point. We find the next point 1 step size later and use the point you just found to approximate.
d/dx a^u
a^u lna
Integral Test
an = f(n) is continuous, positive, and decreasing on [a,∞) if ∫(a,∞)f(n) converges, then the series converges
indefinite integral
an expression, antiderivative + C
Integral
area under the curve
Areas in a Plane definition
areas between curves
lim x->0 sinbx/x
b
lim x->∞ pow top = pow bottom
coefficients
d/dx sinx
cosx
Local Extrema
critical points
Chain Rule
d/dx f(g(x)) = f'(g(x)) g'(x)
Quotient Rule
d/dx f(x)/g(x) = g(x)f'(x)-f(x)g'(x)/g(x)^2
Product Rule
d/dx f(x)g(x) = f(x)g'(x) + g(x)f'(x)
Power rule of Derivatives
d/dx x^n = nx^n-1
Fundamental Theorem of Calculus
d/dx ∫(a,x)f(t)dt = f(x)x'
Velocity>0, Velocity < 0
distance increasing, distance decreasing
Exponential Growth Decay
dy/dt = ky y(t) = y(0)e^kt
Logistical Population Growth
dy/dt = ky(1-y/L) = (k/L)y(L-y) y(t) = L/(1 + be^-kt)
Parametrics
dy/dx = (dy/dt)/(dx/dt)
d/dx e^u
e^u
concavity
f''(x) > 0 concave up f''(x) < 0 concave down
MVT
f'(c) = f(b)-f(a)/b-a
Critical Points
f'(x) = 0 or undefined and endpoints
Zeroes on a calculator
graph it, 2nd trace zero, left/right bound
LPG: before/after the POI the ____ is ______
growth rate is increasing/decreasing
Definite integral
has endpoints, numerical value
limits as x approaches infinity
horizontal asymptotes
Fundamental Theorem of Calculus - Integral Evaluation Theorem
if f is continuous on [a,b] and if F is any antiderivative of f on [a,b] then ∫(a,b) f(x)dx = F(b) - F(a)
Extreme Value Theorem (EVT)
if f is continuous on a closed interval [a,b] then f has both a max and a min on it
Mean Value Theorem (MVT)
if f(x) is continuous on [a,b] and differentiable on (a,b) there is at least one point (c) where the slope of the tangent line must equal the slope of the endpoints (secant line)
Squeeze Theorem
if lim x->a h(x) = lim x->a g(x) = L and h(x)<f(x)<g(x), then lim x->a f(x) = L
Nth Term Test
if lim(n→∞) {an} does not = 0 the series diverges (does not prove convergence)
f'(x) > 0, f'(x) < 0
increasing, decreasing
the slope of tangent line is an
instantaneous rate of change
Power Rule of Integration
integral(x^n dx) = (1/n+1)x^(n+1)
Ratio Test
is lim n->∞ |an+1/an| < 1
Discontinuity
jump, hole/removable, ∞, oscillating
Slope of tangent line (lim def.)
lim h->0 f(x+h) - f(x)/h
Definition of a derivative
lim h->0 f(x+h)-f(x)/h
Continuity
lim x->C = lim x->C+ = lim x->C- = f(c)
lopitals rule
lim x->c f(x)/g(x) = 0/0 DO lim x->c f'(x)/g'(x)
LPG: L is the
limiting value/carrying capacity
A in P when the functions in terms of y
limits are y - values right - left if you rotate so y - axis is bottom, should still be top - bottom ∫(c,d) [f(y) - g(y)]
Rules of logs ln(xy) ln(x/y) ln(x^y) ln(e) ln(1) ln(1/x) kln(x) When differentiating what do you need to make sure you do with natural logs?
lnx + lny lnx - lny ylnx 1 0 -lnx lnx^k ln|x|
f'(x): + to -, - to +
max, min
Ratio Test is inconclusive for all
p-series/=1
Use Exponential Growth Decay when
rate of change (derivative) of an amount (y) is proportional (relative) to the amount
d/dx tanx
sec^2x
∫secxtanxdx
secx + C
d/dx secx
secxtanx
Interval of Convergence
set of values for convergence found with the ratio test Test the endpoints to determine exclusive/conclusiveness
Whatever you pick for u (int by parts) should
simplify if you take its derivative
∫cosxdx
sinx + C
Speed (parametrics)
sqrt(dx/dt^2 + dy/dt^2) where dx/dt = horizontal velocity and dy/dt = vertical velocity
Rectangular to Polar
tan(theta) = y/x r^2 = x^2 + y^2
∫sec^2xdx
tanx + C
A MRAM is an overestimate when underestimate when
the function is concave down concave up
A LRAM is an underestimate when overestimate when
the function is increasing decreasing
A RRAM is an overestimate when underestimate when
the function is increasing decreasing
Areas in a Plane Formula
top function - bottom function ∫(a,b)[f(x) - g(x)] where the graph f(x) is above the graph of g(x)
acceleration>0, acceleration<0
velocity increasing, velocity decreasing
Points of Inflection
where concavity changes or f''(x) = zero or undefined
Maclaurin Series : ln(1+x)
x - x^2/2 + x^3/3 - ... + (-1)^n-1x^n/n = ∑n=1 (-1)^n-1x^n/n (-1<x<=1)
Maclaurin Series : tan^-1(x)
x - x^3/3 + x^5/5 - ... + (-1)^nx^(2n+1)/(2n+1) = ∑n=0 (-1)^nx^(2n+1)/(2n+1) (|x| <= 1)
Maclaurin Series : sinx
x - x^3/3! + x^5/5! - ... + (-1)^nx^(2n+1)/(2n+1)! = ∑n=0 (-1)^nx^(2n+1)/(2n+1)! (all real x)
Polar to Rectangular
x = rcos(theta) y = rsin(theta)
if nothing simplifies (int by parts) do it enough times so
you have a duplicate of the original integral
Slope of a Curve (r = f(theta))/derivative in Polar form
{x = f(theta)cos(theta), y = f(theta)sin(theta)} dy/dx = y'(theta)/x'(theta) --> use product rule to derive
Lagrange Error Bound Formula
|Rn(x)| <= |Max[f^(n+1)(z)](x-c)^(n+1)/(n+1)!| where z is between c and x
Alternating Series Error Bound
|S - Sn| =. |Rn| <= |an+1| S = Sum of the series Sn = Partial Sum Rn = Remainder an+1 = next term
Geometric Series
|r|<1 converges to sum t1/1-r or ar^k/1-r
Washer method (Volumes of Revolution)
π∫(a,b)R^2 - r^2 dx - about a horizontal line π∫(c,d)R^2 - r^2 dy - about a vertical line
Disk Method (Volumes of Revolution)
π∫(a,b)R^2 dx - about a horizontal line π∫(c,d)R^2 dy - about a vertical line
Harmonic Series (p = 1)
∑n=1 1/n
lim x->∞ pow top > pow bottom
∞, -∞
A in P when the 2nd/bottom function starts at point a
∫(0,a)f(x) + ∫(a,b)[f(x) - g(x)]
∫(1/x^n)dx (partial fractions)
∫(A/x^1 + B/x^2 + C/x^n)dx
A in P when top and bottom switch after x = b
∫(a,b) [f(x) - g(x)] + ∫(b,c) [g(x) - f(x)]
Total Distance Traveled (Parametrics)
∫(a,b) sqrt(dx/dt^2 + dy/dt^2) dt
Displacement of a particle from a to b is
∫(a,b)V(x)dx
Total distance traveled w/ V(x)
∫(a,b)|V(x)|dx
Integration by parts
∫udv = uv - ∫vdu
(x,y) unit circle
(cos, sin)
Polar coordiantes
(r, theta) r - directed distance from origin to point p theta - directed angle
LPG: Pt of inflection + max growth rate
(t, L/2) and y = L/2
Circle Equation
(x - h)^2 + (y - k)^2 = r^2
Circle formula Semi circle formula
(x-h)^2 + (y-k)^2 = r^2 where (h, k) is the center of circle with radius r y = +/- sqrt(r^2 - x^2) ) (both + and - make a whole circle)
d/dx cot-1
-1/1+u^2
d/dx cos-1
-1/sqrt(1-u^2)
d/dx csc-1
-1/|u|sqrt(u^2-1)
∫sinxdx
-cosx + C
∫csc^2xdx
-cotx + C
d/dx cotx
-csc^2x
∫cscxcotxdx
-cscx + C
d/dx cscx
-cscxcotx
d/dx cosx
-sinx
cos(π/2)
0
lim x->0 1-cosx/x
0
lim x->∞ pow top < pow bottom
0
sin(0)
0
cos(0)
1
lim x->0 sinx/x
1
sin(π/2)
1
Maclaurin Series : 1/1-x
1 + x + x^2 + ... + x^n = ∑n=0 x^n (|x| < 1)
Maclaurin Series : e^x
1 + x + x^2/2! + ... + x^n/n! = ∑n=0 x^n/n! (all real x)
Maclaurin Series : 1/1+x
1 - x + x^2 - ... + (-x)^n = ∑n=0 (-1)^nx^n (|x|<1)
Maclaurin Series : cosx
1 - x^2/2! + x^4/4! - ... + (-1)^nx^2n/(2n)! = ∑n=0 (-1)^nx^2n/(2n)! (all real x)
Absolute/Conditional Convergence - Alternating Series
1)Converges Absolutely if ∑n=1 |an| converges, then an also converges 2)Converges Conditionally if ∑n=1 |an| diverges but ∑n=1 an converges 3)Diverges Both ∑n=1 |an| and ∑n=1 an diverge
cos - 0, π/6, π/4, π/3, π/2, sin - ""
1, sqrt(3)/2, 1/sqrt(2), sqrt(2)/2, 1/2, 0 the same but reverse order, ascending instead: 0, 1/2, sqrt(2)/2, sqrt(3)/2, 1
How to solve a related rates problem
1. Draw a picture and name the variables and constants 2. Write down additional numerical information 3. Write down what we are to find 4. Write down an equation that relates the variables 5. Differentiate with respect to t(time) - implicitly! 6. Evaluate using numerical information 7. Interpret answer
Power series can converge:
1. to an interval where the radius is the distance from the center to the edge of the interval if. 2. to all real numbers 3. to the center (x = c) only lim n->∞ |an+1/an|... < 1 (1) = 0 (2) = ∞ (3)
Average value theorem
1/(b-a) times the integral on (a, b) of f(x)dx
d/dx loga(u)
1/(lna)u
d/dx tan-1
1/1+u^2
Area bounded by a single polar curve by two polar curves
1/2∫(a,B)r^2 dtheta 1/2∫(a,B)(r1^2 - r2^2) dtheta
derivative of inverse f(a)
1/f'(b)
d/dx sin-1
1/sqrt(1-u^2)
d/dx lnu
1/u
d/dx sec-1
1/|u|sqrt(u^2-1)
Maclaurin Polynomial
A Maclaurin Polynomial is a Taylor Polynomial centered about x = 0. It can be modeled by Pn(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... + f^(n)(0)x^n/n!
Speed at a point
Absolute value of the velocity
Linearization (process)
Approximate a, find the derivative at the approximate, plug into L(x), solve for L(a-original)
antiderivative of 0
Constant C
Limit Comparison Test
Converges if Pick bn to compare to an. Does lim n->∞ an/bn = c, where c > 0 and finite and an, bn > 0 bn converges
Alternating Series Test an = (-1)^nbn
Converges if bn >= 0, function decreasing, lim n->∞ bn = 0
P-Series Test (1/n^p)
Converges if p > 1 Diverges if 0 < p <= 1
Implicit Derivation
Derive with respect to x and y, treating both as functions. Then solve for y prime, which will be present because when you derived the y term, you multiplied it by y prime.