AP Calc ab study

Based off of Dr.Schrieber's "stuff you must know cold..." packet

Alternate definition of the derivative

f’(c) = lim from x→x (f(x)-f(c))/(x-c)

d/dx (x^n)

nx^n-1

d/dx (sin x)

cos x

d/dx (cos x)

-sin x

d/dx (tan x)

sec²x

d/dx (cot x)

-csc²x

d/dx (sec x)

(sec x)(tan x)

d/dx (csc x )

(-csc x)(cot x)

d/dx (ln u)

(1/u)(du/dx)

d/dx (e^u)

(e^u)(du/dx)

Chain rule

d/dx[f(u)] =f’(u)(du/dx) or dy/dx = (dy/du)(du/dx)

Product Rule

d/dx(uv)=u’v+uv’

Quotient Rule

d/dx(u/v) = (u’v-uv’)/v²

Intermediate Value Theorem (IVT)

if f(x) is continuous on [a,b] and y is a number between f(a) and f(b), then there exists at least one number x = c in the open interval (a , b) such that f(c)=y

Mean Value Theorem (MVT)

If the function f(x) is continuous on [a , b] AND first derivative exists on the interval (a, b) then there is at least one number x = cin (a, b) such that f’(c ) = (f(b)-f(a))/(b-a)

Rolle’s Theorem

If the function f(x) is continuos on [a, b] AND the first derivative exists on the interval (a, b) AND f(a) = f(b), then there is at least one number x = c in (a, b) such that f’(c) = (f(b)-f(a))/(b-a)

Extreme Value Theorem (EVT)

If the function f(x) is continuous on [a, b], then the function is guaranteed to have an absolute minimum on the interval

Derivative of an inverst function

If f has an inverse function g then g’(x)' = 1/f’(g(x)) derivatives are reciprocal slopes

Implicit Differentiation

In implicit differentiation you will have a dy/dx for each y in the original function or equation. Isolate the dy/dx.

Average Rate of Change ARoC

m sub sec = (f(b)-f(a))/b-a

Instantaneous Rate of Change IRoC

m sub tan = f’(x) = lim h→0 (f(x+h) - f(x))/h

Critical point

dy/dx = 0 OR undefined, pay attention to endpoints they are not a critical point

Local Minimum

dy/dx goes (-,0,+) or (-, und, +) or d²y/dx²>0

Local Maximum

dy/dx goes (+,0, - ) or (+,und, - ) or (d²y/dx²)<0

Point of inflection

When concavity changes; (d²y/dx²) goes from (+,0, -), (-, 0, +), (+, und, -), or (-,und, +)

First derivative; when f’(x) > 0 the function is…

increasing

First derivative; when f’(x) < 0 the function is…

decreasing

First derivative; when f’(x) = 0 or DNE, critical values exist at…

x

First derivative; Relative Maximum exists at…

f’(x) = 0 or DNE and sign of f’(x) changes from + to -

First derivative; Relative Minimum exists at…

f’(x) = 0 or DNE and sign of f’(x) changes from - to +

Absolute maximum or minimum include endpoints (true or false)

True, both require you check endpoints. Also maximum value is a y-value

Second derivative; when f’’(x) > 0 the function is…

concave up :)

Second derivative; when f’’(x) < 0 the function is…

concave down :(

If f’(x) and sign of f’’(x) changes, there is what at x?

a point of inflection

Second derivative; Relative Maximum exists when…

f’’(x) < 0

Second derivative; Relative Minimum exists when…

f’’(x) > 0

How to write the equation of a tangent line at a point

y2 - y1 = m (x2 - x1); needs slope (derivative) and a point

Horizontal Asymptotes; If the largest exponent in the numerator is < the largest exponent in the denominator then….

the lim (when x is approaching + or - ∞) f(x) = 0

Horizontal asymptotes; If the largest exponnent in the numerator is > the largest exponent in the denominator then…

The lim (when x is approaching + or - ∞) f(x) = DNE

Horizontal asymptotes; If the largest exponent in the numerator is = to the largest exponent in the denominator then…

the quotient of the leading coefficients is the asymptote; lim (when x is approaching + or - ∞) f(x) = a/b

Distance, velocity, and acceleration; x(t) = …

position function; integral of velocity (v(t))

Distance, velocity, and acceleration; v(t) = …

velocity function; the first derivative of position (x(t)) and the integral of acceleration (a(t))

Distance, velocity, and acceleration; a(t) = …

acceleration fuction; the second derivative of position (x(t)) and first derivative of velocity (v(t))

Speed is the absolute value of…?

velocity

Distance, velocity, and acceleration; if acceleration and velocity have the same sign then speed is…?

increasing

Distance, velocity, and acceleration; if acceleration and velocity have different signs then speed is…?

Decreasing

Distance, velocity, and acceleration; if the particle is moving right when velocity is…?

Positive →

Distance, velocity, and acceleration; the particle is moving left when velocity is…?

negative←

Distance, velocity, and acceleration; displacement is…?

the definite integral of velocity

Distance, velocity, and acceleration; total distance is…?

the definate integral from initial time to final time of |v(t)|

Distance, velocity, and acceleration; average velocity is =….?

(final position - initial position)/total time or the change in position (▵x) over total time (▵t)

Distance, velocity, and acceleration; accumulation =…?

x(0) + (the definite integral from t=0 to t=f of v(t))

Logarithms; definition…?

ln N = p < -