Calc terms list

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lim x→a+ 1/x-a

+∞

a-constant/a-very small number

+∞ or -∞

|a| < b

-b<a<b

d/dx cotx

-csc^2x

d/dx cscx

-cscxcotx

d/dx cosx

-sinx

lim x→a- 1/x-a

-∞

d/dx (constant)

0

lim x→0 1-cosx/x

0

lim x→∞ sinx/x

0

limx->∞ sinx/x

0

lim x→0 sinx/x

1

The three reasons that a function is not differentiable at a point are

1) the function is not continuous 2) the graph has a sharp turn 3) the graph has a vertical tangent

process of solving an optimization problem

1. Draw a sketch 2. Write an equation for the item that you want to optimize 3. Write 2nd equation to eliminate a variable by SUBSTITUTION 4. Find the DOMAIN of the function 5. Find the MAXIMUM or MINIMUM A. Use 1st derivative test for open interval B. Use table for closed interval

if f(x) = P(x)/Q(x), with P(x) and Q(x) as polynomial functions, the possible asymptotes are

1. HA: y=o if the degree of numerator is less than the degree of denominator 2. HA: y=the ratio of the leading coefficients, if the numerator and denominator have the same degree. 3. OA: y= the quotient after long division

When substitution yields 0/0 in a limit, you should try

1. factoring 2. expanding 3. common denominator 4. multiply by conjugate 5. one of the 2 known trig limits 6. L'Hopital's rule

Limits fail to exist at a point when

1. the function approaches different values from the left and right 2. the function is unbounded (approaches +∞ or -∞) 3. the function oscillates

sinxcosx

1/2sin2x

sin(2x)=

2sinxcosx

extreme value theorem

If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.

The average rate of change is

The slope between two points or the slope of the secant line

The instantaneous rate of change is

The slope of the tangent line at a single point

The derivative is the slope of

The tangent line

Theorem: If f has a relative minimum or relative maximum at x=c, then c is

a critical number of f

|a|=b when

a=b or -a=b

|a| >b when

a>b or a<-b

the derivative of the velocity is

acceleration

the second derivative of the position is

acceleration

holes in a graph occur when

both the numerator and denominator are equal to zero and there is not a vertical asymptote

The restriction on the domain of a/b is

b≠0

cos(2x)

cos^2x-sin^2x 2cos^2x-1 1-2sin^2x

d/dx sinx

cosx

cos(x+-y)

cosxcosy-+sinxsiny

the chain rule states that

d/dx[f(u)] =f'(u)u'

f(x) is concave down when f' is________ or f'' is _______

decreasing, f''<0

vertical asymptotes occur when

denominator=0, but numerator ≠ 0

The slopes of 2 parallel lines are

equal

If f is differentiable at a point c, then

f is continuous at x=c

By the 1st derivative test a point has a relative minimum when

f' changes from a negative to a positive

By the 1st derivative test a point has a relative maximum when

f' changes from a positive to negative

The alternate definition of the derivative at a single point is

f'(c)= limx->c f(x)-f(c)/x-c

By the 2nd derivative test, a point is a relative maximum at x=c if

f'(c)=0 and f''<0

By the 2nd derivative test, a point is a relative minimum at x=c if

f'(c)=0 and f''>0

The definition of the derivative is

f'(x) = lim h->0 f(x+h) - f(x)/h

d/dx [f+g]

f'+g'

f(x) is decreasing when

f'<0

f(x) has a critical point when

f'=0 or f' is undefined

f(x) is increasing when

f'>0

d/dx [f*g]

f'g +fg'

in order to approximate using differentials, f(x+∆x)=

f(x) + f'(x)*∆x

d/dx [f/g]

gf'-fg'/g^2

rolles theorem says

if f(a)=f(b), f(x) is differentiable on (a,b) and f(x) is continuous on [a,b], then f'(c)=0 for c ∊ (a,b)

the mean value theorem says

if f(x) is differentiable on (a,b) and continuous on [a,b], then f'(c)= f(b)-f(a)/b-a for c ∊ (a,b)

f(x) is concave up when f' is________ or f'' is _______

increasing, f''>0

horizontal asymptotes occur when

lim x→+-∞ f(x)=L. the asymptote is y=L

the slopes of 2 normal lines are

negative reciprocals

0(nonzero/0) is

not equal to 0, you must simplify to find the limit

d/dx u^n

nu^(n-1)u'

d/dx (x^n)

nx^n-1

The equation of motion for a free-falling object under the force of gravity is

s(t)=1/2gt^2+v0t+s0

d/dx tanx

sec^2x

d/dx secx

secxtanx

a/b > 0 when

signs of a and b are the same

sin(x+-y)

sinxcosy+-sinycosx

in a rational expression when the power of the numerator exceeds the power of the denominator by one, there is a

slant asymptote found by dividing the expression

A function is odd if

symmetric to origin, or when the point (a,b) is on the graph, so is the point (-a,-b), or f(-x) = -f(x)

A function is even if

symmetric to y-axis, or when the point (a,b) is on the graph, so is the point (-a,b), or f(-x) = f(x)

f(x) has an inflection point at (c, f(c)) when

the concavity changes at f''(c)=0 or f''(c) is undefined

In order to check for an absolute extrema on a closed interval you must check

the critical points and the endpoints on the interval

vertical asymptotes occur when

the denominator equals 0, but the numerator is not equal to zero

horizontal tangents occur when

the derivative is equal to 0

In a rational expression, when the power of the denominator equals the power of the numerator, there is a horizontal asymptote at

the ratio of the leading coefficients of the numerator and the denominator

The derivative of the position is

velocity

Graphically a parabola has a maximum or minimum at its vertex where x=

x=-b/2a

The restriction on the domain of lnx is

x>0

the restriction on the domain of √x is

x≥0

The equation of a line with a slope m passing through (a.b) is

y-b=m(x-a)

In a rational expression, when the power of the denominator exceeds the power of the numerator, there is a horizontal asymptote at

y=0

The equation of a line with slope m and y-intercept b is

y=mx+b


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