Calc terms list
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