EXAM 1 - MATH 221

memorize

f(x)=e^x … f’(x)=?

f’(x)=e^x

f(x)=ln(x)…f’(x)=?

f’(x)=(1/x)

f(x)=sin(x)…f’(x)=?

f’(x)= cos(x)

f(x)= cos(x) … f’(x)=?

f’(x)= -sin(x)

f(x)= tan(x) … f’(x)=?

f’(x)= sec²(x)

f(x)= cot(x) … f’(x)=?

f’(x)= -csc²(x)

f(x)= sec(x) … f’(x)=?

f’(x)= sec(x)tan(x)

f(x)= csc(x) … f”(x)=?

f’(x)= -csc(x)cot(x)

lim(x—>a) f(x) = L

If f(x) is defined on some open interval around a (but NOT equal to a) we can force f(x) as close as we like to L by choosing x close enough to a. This is the formal definition of the limit of f(x) as x approaches a.

delta epsilon def

For every epsilon>0 there exists a delta>0 so that if x is in (a-delta, a+delta) then f(x) is in (L - epsilon, L + epsilon)

Squeeze theorem

(BY SQZ THRM) if f(x) <= g(x) <= h(x) for x near (NOT @ A) and lim(x—>a) f(x) =L = lim(x—>a) , then lim(a—>a) g(x)=L

F is continuous @ a if

1. function is defined

2. limit exists

3. lim(x→a)f(s) = f(a)

• 1 & 2 must match

Intermediate Value Theorem

If f is continuous on [a,b] and N is a number between f(a) & f(b) then there exists some C in (a,b) where f( c ) = N

formal def of derivative

f’(a) = lim(h→0) (f(a+h) - f(a))/h

power rule

For a function f(x) = x^n, the derivative f'(x) is given by n*x^(n-1).

sum rule

The rule stating that the derivative of the sum of two functions is the sum of their derivatives. If f and g are differentiable functions, then (f + g)' = f' + g'.

product rule

The rule that states the derivative of a product of two functions is given by (f * g)' = f' * g + f * g'.

quotient rule

The rule that states the derivative of a quotient of two functions is given by (f / g)' = (f' * g - f * g') / g^2, where g is not equal to zero.

chain rule

The rule used to compute the derivative of a composite function, stating that if a function y = f(g(x)), then the derivative is given by y' = f'(g(x)) (g'(x)).