Derivatives
Limit Definition of a Derivative (in terms of delta x or h)
lim_(Deltax->0) [f(Deltax+x)-f(x)]/(Deltax)
lim_(h->0) [f(h+x)-f(x)]/h
int_a^bf(u)du=?
Hint: (First FTC)!!
int_a^bf(u)du=F(b)-F(a)
Explain IVT
Answers will vary but should include:
The function is continuous & differentiable on the closed interval [a,b].
Used to prove that value f(c) exists on the interval [a,b], can be used to prove a zero occurs.y=2sin2x-8tanxsecx,
y'=?
4cos2x-8secx(sec^2x+tan^2x)
d/dxarccosu=?
d/dxarccosu=(-u')/sqrt(1-u^2)
d/dxcu=?
d/dxcu=cu'
int_a^af(x)dx=?
int_a^af(x)dx=0
Rolle's Theorem
Answers will vary but should include information similar to:
If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and f(a) = f(b), then there exists a number c in (a,b) such that f'(c) = 0
int(secutanu)du=?
int(secutanu)du=secu+c
d/dxarcsecu=?
d/dxarcsecu=(u')/[abs(u)sqrt(u^2-1)]
g'(x)=?
g'(x)=1/[f'(g(x))
int 1/(u)du=?
int 1/udu=lnabs(u)+c
Explain MVT
Answers will vary but should include:
The function must be continuous and differentiable on the open interval (a,b).
int (sin^2x+cos^2x)/(cot^2xsec^2x)
int (sin^2x+cos^2x)/(cot^2xsec^2x)=-cotx+c
d/dxsin^-1u=?
d/dxsin^-1u=(u')/sqrt(1-u^2)
The Power Rule in Terms of u
d/dxu^n=n(u)^(n-1)u'
int e^udu=?
int e^udu=e^u+c
First Fundamental Theorem of Calculus
int_a^b f(u)du=F(b)-F(a)
sin^2y+8sinx+4cos5x=90
dy/dx=? Use Implicit Differentiation.
dy/dx=-(8cosx-20sin5x)/(2sinycosy)
d/dxcot^-1u=?
d/dxcot^-1u=(-u')/(u^2+1)
d/dxe^u=?
e^u u'
inta^udu=?
inta^udu=1/(lna)a^u+c
Second Fundamental Theorem of Calculus
d/dtint_a^xf(t)dt=f(x)
int (cscu)du=?
intcscudu=-lnabs(cscu+cotu)+c
d/dxarctanu=?
d/dxarctanu=(u')/(1-u^2)