Limits
f(x) = x + 1
The limit as x approaches 1
2
f'(x) is equal to
3
A function f(x) is continuous on a closed interval [a,b] takes on every value between f(a) and f(b).
Intermediate Value Theorem
When f '(x) is positive, f(x) is
increasing
f(x) = 2x + 4x + 6
The limit as x approaches 0.5
9
f'(x) is equal to
14x
If f is continuous on [a,b] and differentiable on (a,b) then there is at least one number c such that [f(a) - f (b)] / [a - b] = f'(c).
Mean Value Theorem
When f '(x) changes from positive to negative, f(x) has a
relative maximum
f(x) = [sin(2x)] / x
The limit as x approaches 0
2
f(x) = sin(x)
f'(x) is equal to
cos(x)
If f is continuous on [a,b] then f(x) has both a maximum and minimum on [a,b].
Extreme Value Theorem
When f '(x) is decreasing, f(x) is
concave down
f(x) = [1 - cos(x)] / x
The limit as x approaches 0
0
f(x) = tan(x)
f'(x) is equal to
sec2(X)
If f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.
Squeeze Theorem
When f ''(x) changes from increasing to decreasing or decreasing to increasing, f(x) has a
point of inflection
f(x) = (x-5) / [(x-5)(x+5)]
The limit as x approaches 5
1/10
f'(x) is equal to
8x3 + 9x2 + 3
The integral on (a, b) of f(x) dx = F(b) - F(a)
Fundamental Theorem of Calculus
When velocity and acceleration are the same sign, the particle is
speeding up