84 terms

# AP BC Calculus: Proficiency Formulas (S1)

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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.
Average Rate of Change
Slope of secant line between two points, use to estimate instantanous rate of change at a point.
Instantenous Rate of Change
Slope of tangent line at a point, value of derivative at a point
Formal definition of derivative
limit as h approaches 0 of [f(a+h)-f(a)]/h
Alternate definition of derivative
limit as x approaches a of [f(x)-f(a)]/(x-a)
When f '(x) is positive, f(x) is
increasing
When f '(x) is negative, f(x) is
decreasing
When f '(x) changes from negative to positive, f(x) has a
relative minimum
When f '(x) changes fro positive to negative, f(x) has a
relative maximum
When f '(x) is increasing, f(x) is
concave up
When f '(x) is decreasing, f(x) is
concave down
When f '(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
point of inflection
When is a function not differentiable
corner, cusp, vertical tangent, discontinuity
Product Rule
uv' + vu'
Quotient Rule
(uv'-vu')/v²
Chain Rule
f '(g(x)) g'(x)
Particle is moving to the right/up
velocity is positive
Particle is moving to the left/down
velocity is negative
absolute value of velocity
speed
y = sin(x), y' =
y' = cos(x)
y = cos(x), y' =
y' = -sin(x)
y = tan(x), y' =
y' = sec²(x)
y = csc(x), y' =
y' = -csc(x)cot(x)
y = sec(x), y' =
y' = sec(x)tan(x)
y = cot(x), y' =
y' = -csc²(x)
y = sin⁻¹(x), y' =
y' = 1/√(1 - x²)
y = cos⁻¹(x), y' =
y' = -1/√(1 - x²)
y = tan⁻¹(x), y' =
y' = 1/(1 + x²)
y = cot⁻¹(x), y' =
y' = -1/(1 + x²)
y = e^x, y' =
y' = e^x
y = a^x, y' =
y' = a^x ln(a)
y = ln(x), y' =
y' = 1/x
y = log (base a) x, y' =
y' = 1/(x lna)
To find absolute maximum on closed interval [a, b], you must consider...
critical points and endpoints
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)
If f '(x) = 0 and f"(x) > 0,
f(x) has a relative minimum
If f '(x) = 0 and f"(x) < 0,
f(x) has a relative maximum
Linearization
use tangent line to approximate values of the function
left riemann sum
use rectangles with left-endpoints to evaluate integral (estimate area)
right riemann sum
use rectangles with right-endpoints to evaluate integrals (estimate area)
trapezoidal rule
use trapezoids to evaluate integrals (estimate area)
average value of f(x)
= 1/(b-a) ∫ f(x) dx on interval a to b
If g(x) = ∫ f(t) dt on interval 2 to x, then g'(x) =
g'(x) = f(x)
Fundamental Theorem of Calculus
∫ f(x) dx on interval a to b = F(b) - F(a)
To find particular solution to differential equation, dy/dx = x/y
separate variables, integrate + C, use initial condition to find C, solve for y
To draw a slope field,
plug (x,y) coordinates into differential equation, draw short segments representing slope at each point
methods of integration
substitution, parts, partial fractions
∫ u dv =
uv - ∫ v du
dP/dt = kP(M - P)
logistic differential equation, M = carrying capacity
P = M / (1 + Ae^(-Mkt))
logistic growth equation
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
volume of solid of revolution - no washer
π ∫ r² dx over interval a to b, where r = distance from curve to axis of revolution
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
length of curve
∫ √(1 + (dy/dx)²) dx over interval a to b
L'Hopitals rule
use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit
indeterminate forms
0/0, ∞/∞, ∞*0, ∞ - ∞, 1^∞, 0⁰, ∞⁰
derivative of parametrically defined curve
x(t) and y(t)
dy/dx = dy/dt / dx/dt
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
length of parametric curve
∫ √ (dx/dt)² + (dy/dt)² over interval from a to b
given velocity vectors dx/dt and dy/dt, find speed
√(dx/dt)² + (dy/dt)² not an integral!
given velocity vectors dx/dt and dy/dt, find total distance travelled
∫ √ (dx/dt)² + (dy/dt)² over interval from 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
area inside one polar curve and outside another polar curve
1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.
Definition of Continuity
A function is continuous if 1) f(c) is defined. 2) lim f(x) as x approaches c exists 3) lim f(x) as x approached c equals f(c)
d/dx(arcsecx)=
y'= x'/(I x I*√(x² - 1))
Fundamental Theorem of Calculus Part 2
if f(x) is continuous on an open interval I, then d/dx[∫f(t)dt] = f(x)
∫sinxdx
-cosx+ C
∫tanxdx
-ln IcosxI + C
∫secxdx
ln Isecx + tanxI + C
∫secxtanxdx
secx + C
∫csc^2xdx
-cotx + C
∫a^xdx
(1/lna)(a^x) + C
∫1/(a^2 + x^2)dx
(1/a)arctanx/a + C
∫cosxdx
sinx + C
∫cotxdx
ln IsinxI + C
∫cscxdx
-ln Icscx + cotxI + C
∫sec^2xdx
tanx + C
∫cscxcotxdx
-cscx + C
∫1/ (a^2 - x^2)dx
arcsin (x/a) + C
sin^2x=
(1-cos2x)/2
cos^2x=
(1+cos2x)/2
Parametric Form of 2nd Derivative
(d/dt(dy/dx))/(dx/dt)
Rolle's Theorem
If f(x) is continuous on the closed interval [a, b], differentiable on (a, b), and satisfies f(a) = f(b), then for some c in the interval (a, b), we have f'(c) = 0
Second Derivative Test
if f'(c) = 0 and f''(c) > 0 then minimum; if f'(c) = 0 and f''(c) < 0 then maximum