Definitions
A function f is continuous a x=c if and only if:
1) f(c) exists
2) the limit (as x approaches c) f(x) exists
3) the limit (as x approaches c) f(x)=f(c)
Definition of Continuity
y=x3sin2x
Find dy/dx
y' = 3x2sin2x + 2x3cos2x
y=(7x-2)/(3x+4)
Find dy/dx
y' = 34/(3x+4)2
y=tanx
y'=sec2x
Name (or List) a Pythagorean Identity
1+cot2x=csc2x
1-csc2x=-cot2x
tan2x=sec2x-1
If a function f is continuous on [a,b] and y is a number between f(a) and f(b), then there exists x=c such that f(c)=y.
Definition of Intermediate Value Theorem
y=5(x2-6x+11)3
Find dy/dx
y' = 15(x2-6x+11)2(2x-6)
or
y' = (30x-90)(x2-6x+11)2
f(x)=(x+2)/(cos3x)
Find f'(x)
f'(x) = (cos3x+3(x+2)(sin3x))/(cos23x)
y=cotx
y'=-csc2x
y=ln(3x2)
Find y'
y'=2/x
If 1) f is differentiable for all values of x in the open interval (a,b) and 2) f is continuous at x=a and x=b, then there is at least one number x=c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a).
Definition of Mean Value Theorem
y=x4cos53x
Find dy/dx
y' = 4x3cos53x - 15x4cos43xsin3x
y=x3/sinx
Find y'
y' = (3x2sinx-x3cosx)/(sin2x)
y=secx
y'=secxtanx
y=exee
Find y'
y'=exee
L is the limit of f(x) as x->c if and only if for any positive number epsilon, there exists a positive number delta such that if x is within delta units of c, but not equal to c, then f(x) is within epsilon units of L.
Definition of Limit
y=(x2-1)10(x2+1)15
Find dy/dxy' = 10x(x2-1)9(x2+1)14(5x2-1)
y=(sin10x)/(cos20x)
Find y'
y' = (10cos20xcos10x+20sin10xsin20x)/(cos220x)
y=(secx)(cotx)
y'=-cscxcotx
y=(e-x+e2x)5
Find y'
y'=5(e-x+e2x)4(-e-x+2e2x)
If f is continuous on the closed interval [a,b], then there are numbers c1 and c2 in [a,b] for which f(c1) and f(c2) are the maximum and minimum values of f(x) for that interval.
Definition of Extreme Value Theorem
y=10cos85xsin58x
Find dy/dx
y' = -400cos75xsin5xsin58x + 400cos85xsin48xcos8x
y=(5x2-10x+3)/(3x2+6x-8)
Find y'
y' = (60x2+182x+62)/((3x2-6x-8))2
y=-3sec23x
y'=-18sec23xtan3x
x(t)=(1/3)t3-2t2+3t+1
Find the velocity and acceleration equationsv(t)=t2-4t+3
a(t)=2t-4