# 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)

increasing

decreasing

relative minimum

relative maximum

concave up

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

uv' + vu'

(uv'-vu')/v²

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

speed

y' = cos(x)

y' = -sin(x)

y' = sec²(x)

### y = csc(x), y' =

y' = -csc(x)cot(x)

### y = sec(x), y' =

y' = sec(x)tan(x)

y' = -csc²(x)

y' = 1/√(1 - x²)

### y = cos⁻¹(x), y' =

y' = -1/√(1 - x²)

y' = 1/(1 + x²)

y' = -1/(1 + x²)

y' = e^x

y' = a^x ln(a)

y' = 1/x

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

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

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)

-cosx+ C

-ln IcosxI + C

### ∫secxdx

ln Isecx + tanxI + C

secx + C

-cotx + C

(1/lna)(a^x) + C

### ∫1/(a^2 + x^2)dx

(1/a)arctanx/a + C

sinx + C

ln IsinxI + C

### ∫cscxdx

-ln Icscx + cotxI + C

tanx + C

-cscx + C

arcsin (x/a) + C

(1-cos2x)/2

(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

Example: