AP Calc BC Formula List

Some parentheses have not been included for the sake of easier typing. The underlined terms are ones confirmed to be on the quiz. The highlighting is purely for distinction of multiple responses.

|x| =

{-x, x<0

x, x>=0

Distance between two points (x1, y1) and (x2,y2)

sqrt((x1-x2)^2+(y1-y2)^2)

The graph of an even function is symmetric with respect to the

y axis, f(-x) = f(x)

The graph of an odd function is symmetric with respect to the

origin, f(-x) = -f(x)

(sinx)^2+(cosx)^2 =

1

1+(tanx)^2 =

(secx)^2

1+(cotx)^2 =

(cscx)^2

sin2x =

2sinxcosx

cos2x =

==(cosx)^2-(sinx)^2== = $1-2(sinx)^2$ = %%2(cosx)^2-1%%

(cosx)^2 =

1+cos2x/2

(sinx)^2 =

1-cos2x/2

Squeeze Theorem: If ____ for all x in an open interval containing c, except possibly at c itself, and if ____ , then ____

==h(x)

lim\[x→0\]sinx/x (keep in mind the 6 other ways this can be written) =

1

lim\[x→0\](1-cosx)/2 =

0

__Definition of Continuity at a Point__

f(c) is defined

lim\[x→c\]f(x) exists, that is lim\[x→c-\]f(x) = lim\[x→c+\]f(x)

lim\[x→c\]f(x) = f(c)

__Definition of Continuity at a Point: COMBINED__

A function f is continuous at x = c iff **lim[x→c-]f(x) = lim[x→c+]f(x) = f(c)**

Removable discontinuity at x = c

point(hole)

Non-removable discontinuity at x = c

jump or vertical asymptote

A function is continuous on an interval if ____

the function is continuous at each point in the interval

A function is continuous everywhere if ____ (check packet for second sentence)

the function is continuous for all real number x.

Intermediate Value Theorem:

If f is ____ on \[a,b\] and k is any number between f(a) and f(b), then there is ____ such that ____.

==continuous==, $at least one number c between a and b$, %%f(c) = k%%

__Average Rate of Change of f(x) on closed interval [a,b]__

f(b)-f(a)/b-a

Definition of the Derivative: OPTION ONE (Δ, for copy-paste purposes)

f’(x) = lim\[Δx→0\]f(x+Δx)-f(x)/Δx

Definition of the Derivative: OPTION TWO

f’(x) = lim\[h→0\]f(x+h)-f(x)/h

Definition of the Derivative: OPTION THREE(Alternative Form)

f’(c) = lim\[x→c\]f(x)-f(c)/x-c

A derivative is {

slope of the tangent line, limit of the slopes of secant lines, instantaneous rate of change, limit of average rates of change

An equation of the line **tangent** to the graph of f(x) at the point (c,f(c)) is

y - f(c) = f’(c)(x-c)

An equation of the line **normal** to the graph of f(x) at the point (c,f(c)) is

y - f(c) = (-1/f’(c))(x-c)

The Constant Rule:

d/dx (c) = 0

The Power Rule(normal):

d/dx (x^n) = nx^(n-1)

The Power Rule(with u):

d/dx (u^n) = nu^(n-1)du/dx

The Constant Multiple Rule:

d/dx (cf(x)) = cf’(x)

The Sum and Difference Rule:

d/dx (f(x) +- g(x)) = f’(x) +- g’(x)

The Product Rule:

d/dx (f(x)g(x)) = f(x)g’(x)+f’(x)g(x)

The Quotient Rule:

d/dx (f(x)/g(x)) = g(x)f’(x)-f(x)g’(x)/(g(x))^2

The Chain Rule(normal):

d/dx (f(g(x))) = f’(g(x))\*g’(x)

The Chain Rule(with u):

d/dx ((u(x))^n) = n(u(x))^n-1\*u’(x)

d/dx (sinx) =

cosx

d/dx (cosx) =

\-sinx

d/dx (tanx) =

(secx)^2

d/dx (cotx) =

\-(cscx)^2

d/dx (secx) =

secxtanx

d/dx (cscx) =

\-cscxcotx

d/dx (lnx) =

1/x

d/dx (log\[a\]x) =

1/xlna

d/dx (e^x) =

e^x

d/dx (a^x) =

a^x\*lna

d/dx (arcsinx) =

1/sqrt(1-x^2)

d/dx (arccosx) =

\-1/sqrt(1-x^2)

d/dx (arctanx) =

1/1+x^2

d/dx (arccotx) =

\-1/1+x^2

d/dx (arcsecx) =

1/|x|sqrt(x^2-1)

d/dx (arccscx) =

\-1/|x|sqrt(x^2-1)

__(f^-1)’(a) =__

1/f’(f^-1(a))

Definition of Extrema: Let f be ____ on an interval I containing c

defined

f(c) is the maximum of f on I if f(c) >= f(x) for all x in I

f(c) is the minimum of f on I if f(c)

Definition of Relative Maximum: If there is an ____ containing c on which ____, then f(c) is a relative maximum.

==open interval==, $f(c) >= f(x) for all x in that interval$

Definition of Relative Minimum: If there is an ____ containing c on which ____, then f(c) is a relative minimum.

==open interval==, $f(c) <= f(x) for all x in that interval$

Extreme Value Theorem: If f is ____ on a closed interval \[a,b\] then ____

continuous, f has both a maximum and a minimum on \[a,b\]

The Closed Interval Test(The Candidates Test): PART ONE

Find the critical numbers of f in (a,b)

The Closed Interval Test(The Candidates Test): PART TWO

Evaluate f at each critical number in (a,b)

The Closed Interval Test(The Candidates Test): PART THREE

Evaluate f at each end point of \[a,b\]

The Closed Interval Test(The Candidates Test): PART FOUR

The least of these values is the minimum. The greatest is the maximum.

Rolle’s Theorem: If f is ____ on \[a,b\] and ____ on (a,b) and if ____, then there is at least one number c on (a,b) such that ____.

==continuous==, $differentiable$, %%f(a) = f(b)%%, ^^f’(c) = 0^^

Mean Value Theorem: If f is ____ on \[a,b\] and ____ on (a,b), then there exists a number c on (a,b) such that ____.

==continuous==, $differentiable$, ^^f’(c) = f(b)-f(a)/b-a^^

Definition of a Critical Number: Let f be ____ at c. If ____.

==defined==, $f’(c) = 0 or if f’ is undefined at c, then c is a critical number of f$

First Derivative Test: Let c be a ____ of a function f that is ____ on an open interval I containing c. If f is ____ on the interval, except possibly at c, then ____ ____.

==critical number==, $continuous$, %%differentiable%%, ^^If f’ changes from negative to positive at c, then f(c) is a relative minimum of f^^, @@If f’ changes from positive to negative at c, then f(c) is a relative maximum of f@@

Second Derivative Test: Let f be a function such that ____ and ____ on an open interval containing c. If ____. If ____. If ____.

==f’(c) = 0==, $f’’ exists$, %%f’’(c) > 0, then f(c) is a relative minimum%%, ^^f’’(c) < 0, then f(c) is a relative maximum^^, @@f’’(c) = 0, then the Second Derivative Test fails, use the First Derivative Test@@

Definition of Concavity: Let f be ____ on an open interval I. The graph of f is ____ on I if ____ and ____ on I if ____ .

==differentiable==, $concave upward$, %%f’ is increasing on the interval%%, ^^concave downward^^, @@f’ is decreasing on the interval@@

Test for Concavity: Let f be a function whose ____ exists on an open interval I. If ____. If ____.

==second derivative==, $f’’(x) > 0 for all x in I, then the graph of f is concave upward in I$, %%f’’(x) < 0 for all x in I, then the graph of f is concave downward in I%%

Definition of a Point of Inflection: A function f has a point of inflection at (c,f(c))

(1) ____

(2) ____

==if f’’(c) = 0 or f’’(c) does not exist and==, $if f’’ changes sign from positive to negative or negative to positive at x = c or if f’(x) changes from increasing to decreasing or decreasing to increasing at x = c.$

__Definition of a Definite Integral__: A definite integral is the limit of a Riemann sum.

memorize

∫0dx =

C

∫kdx =

kx + C

∫dx =

x + C

∫((f(x)+-g(x))dx =

∫f(x)dx +- ∫g(x)dx

∫x^ndx =

1/n+1\*x^n+1 + C, n!= -1

∫cosxdx =

sinx + C

∫sinxdx =

\-cosx + C

∫(secx)^2dx =

tanx + C

∫(cscx)^2dx =

\-cotx + C

∫secxtanxdx =

secx + C

∫cscxcotxdx =

\-cscx + C

∫1/xdx =

ln|x| + C

∫tanxdx =

\-ln|cosx| + C

∫cotxdx =

ln|sinx| + C

∫secxdx =

ln|secx+tanx| + C

∫cscxdx =

\-ln|cscx+cotx| + C

∫e^xdx

e^x + C

∫a^xdx

a^x/lna + C

∫dx/sqrt(a^2-x^2) =

arcsin(x/a) + C

∫dx/x^2+a^2 =

1/a\*arctan(x/a) + C

∫dx/x\*sqrt(x^2-a^2) =

1/a\*arcsec(|x|/a) + C

Fundamental Theorem of Calculus:

∫\[a-b\]f’(x)dx = f(b)-f(a)

Second Fundamental Theorem of Calculus:

d/dx∫\[a-x\]f(t)dt = f(x)

Chain Rule Version of Second FTC:

d/dx∫\[a-g(x)\]f(t)dt = f(g(x))\*g’(x)

Average Value of f(x) on Closed Interval \[a,b\] =

∫\[a-b\]f(x)dx/b-a

Definition of e:

lim\[n→inf\](1+1/n)^n

Volume around a Horizontal Axis by Disks:

V=pi∫\[a-b\](r(x))^2dx

Volume around a Vertical Axis by Disks:

V=pi∫\[a-b\](r(y))^2dx

Volume around a Horizontal Axis by Washers:

V=pi∫\[a-b\](R(x))^2-(r(x))^2dx