CALC COLD QUIZ

derivative of an inverse function

(f^-1)'(a)= 1/f'(f^-1(a)) or if f(g(x))=x then g'(x)= 1/f'(g(x))

d/dx (csc u)

-csc(u)cot(u) du/dx

d/dx (cot u)

-csc^2 u du/dx

d/dx (cos u)

-sin(u) du/dx

Finding absolute Global Max/Min

Absolute extrema on a closed interval is only located at critical numbers/endpoints. This is known as the Candidates Test

IVT

If f is continuous on [a, b] and differentiable on (a, b) then there exists a number c on (a, b) such that f'(c)= f(b)-f(a)/b-a

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b].

Concavity

If f is differentiable, the graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

Inflection point meaning

Inflection point (c, f(c)) 1. If f''(c)=0 or f''(c) DNE and... 2. if f'' changes sign between positive/negative at x=c OR if f'(x) changes between increasing/decreasing at x=c

First Derivative Test

Let c be a critical number of a continuous function f on an open interval I containing c. If f is differentiable on the interval, except at x=c then f(c) can be either: 1. relative minimum --> neg to pos 2. relative maximum --> pos to neg

Test for increasing/decreasing functions

Let f be a continuous function on the closed interval (a,b) 1. If f'(x)>0 for all x in (a,b), then f is increasing on [a,b] 2. If f'(x)<0 for all x in (a,b), then f is decreasing on [a,b] 3. If f'(x)=0 for all x in (a,b), then f is constant on [a,b]

Second Derivative Test

Let f be a function such that the second deruvative of f exists on an open interval of c. 1. If f'(c)=0 and f''(c)>0, then (c, f(c)) is a rel min of f 2. If f'(c)=0 and f''(c)<0, then (c, f(c)) is a rel max of f

Test for concavity

Let f be a function whose second derivative exists on an open interval I. 1. If f''(x)>0 for all x in the interval I, then the graph of f is concave upward in I 2. If f''(x)<0 for all x in the interval I, then graph of f is concave downward in I

Definition of a critical number

Let f be defined at c. If f'(c)=0 or if f' is undefined at c, then c is a critical number of f

d/dx (sin u)

cos(u) du/dx

Definition of increasing and Decreasing functions

f is increasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2, implies fx1<x2 f is decreasing on an interval if for any 2 numbers x1 and x2 in the interval x1<x2 implies fx1>fx2

alternate form definition of derivative

f'(c)=lim x->c f(x)-f(c)/x-c

d/dx [f(g(x))]

f'(g(x))g'(x)

d/dx [f(x)*g(x)]

f'(x)*g(x) + f(x)g'(x) PAY ATTENTION TO SYMBOLS!!!

limit definition of derivative

f'(x)= lim h->0 f(x+h)-f(x)/h

Average Rate of Change of f(x) on [a,b]

f(a)-f(b)/a-b

point slope form of a line (a)

f(x)-f(a)=f'(a)(x-a)

d/dx[x^n]

nx^(n-1)

d/dx (tan u)

sec^2(u) du/dx

d/dx (sec u)

secutanu du/dx