Limits
(x->c = as x is approaching c)
(x->c = as x is approaching c)
What makes a function continuous?
The function must be defined at x=c
The limit of the function as x approaches c must exist
The limit must equal the function's value at x=c
What is a derivative?
The slope of the tangent line to the graph
What is an integral?
The area under the curve/function
cos(x)
What theorem does this illustrate?
f(x) </= g(x) </= h(x)
lim(x->c) f(x) = lim(x->c) h(x) = L
lim(x->c) g(x) = L
Squeeze Theorem
Evaluate:
lim(x->-1) lxl
1
What is d/dx (1,234,567x) ?
1,234,567
What is ~ x2
x3/3 + c
What is the integral of sec(x)tan(x) ?
What theorem does this illustrate?
L(x) = a~x f(t) dt
is continuous on [a,b] and differentiable on (a,b)
L'(x) = d/dx [ a~x f(t) dt ] = f(x)
for all x in (a,b)
Fundamental Theorem of Calculus
Evaluate:
lim(x->5) x2+2x-4
31
What is d/dx (4x3) ?
12x2
What is ~ 5/x
5lnx
What is the derivative of csc(x) ?
-csc(x)cot(x)
What rule does this illustrate?
F'(x) = f'[g(x)] * g'(x)
Chain Rule
Evaluate:
lim(x->2) (x2-4)/(x-2)
4
What is d/dx lnx ?
1/x
What is ~ 3x dx
3x/ln3 + c
1/(1+x2)
What theorem does this illustrate?
if f and g are continuous on [a,b] and differentiable on (a,b)
AND if g'(x) does not equal 0 on (a,b)
then there is a number c in (a,b) for which
f'(c)/g'(c) = f(b)-f(a) / g(b)-g(a)
Mean Value Theorem
Evaluate:
lim(x->3) (x2-9)/(x2-x-6)
6/5
What is the derivative of a function that demonstrates a particle's velocity?
Its acceleration
What is the integral of a function that demonstrates a particle's velocity?
Its position
Evaluate:
0~π/2 10sin(x) dx
(integral from π to π/2 of 10sin(x)) (π = pi)
10
What rule does this illustrate?
f(x)/g(x) is indeterminate form at c (0/0)
if the lim(x->c) f'(x)/g'(x) = L
then lim(x->c) f(x)/g(x) = L
L'Hôpital's Rule