Memorization Quiz 1

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

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

A function does not have a limit at x=a if

The limit approaching f(x) from the left and right is not equal

The function is oscillating at x=a

The function isn't nearing a finite value at x=a

A function has a limit at x=a if and only if

the lim f(x) as x approaches a from the left and the right are equal

ln e

1

Ln 1

0

sin^2theta+cos^2theta=

1

The Squeeze Theorem

If h(x) ≤ f(x) ≤ g(x) anf if lim h(x) = lim g(x) = L, than lim f(x) = L

A function is continuous at x=a if

lim f(x) from the left and the right = f(a)

The Intermediate Value Theorem

If f(x) is continuous on [a,b] and k is any number between f(a) and f(b), then there exists c in (a,b) such that f(c)=k

Give an equation using limits that represents f`x for all x

f`(x) = lim as h approches 0 f(x+h) - f(x)/h

the derivative of a function is also referred to as the ______ of that function

Instantaneous rate of change

Give two equations using limits to represent f`(a)

f`(x) = lim as h approches 0 f(x+h) - f(x)/h

f`(a) = lim as x approaches a f(x) - f(a)/x-a

f has a _______ at x=c if f`(c) changes from negative to positive at x = c

relative minimum

f has a _______ at x=c if f`(c) changes from positive to negative at x = c

relative maximum

On an interval where f` < 0, f is

decreasing

On an interval where f` > 0, f is

increasing

A function f(x) has a derivative @ x=a if

the limit from the left = limit from the right = f(a)

the derivative from the left = the derivative from the right

d/dx c

0

d/dx x^n

nx^n-1

d/dx [af(x)]

af`(x)

d/dx sinx

cosx

d/dx cosx

-sinx

d/dx e^x

e^x

d/dx lnx

1/x

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

f`(x)g(x) + f(x)g`(x)

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

g(x)f`(x) - f(x)g`(x)/g(x)^2

d/dx tanx

sec^2x

d/dx cotx

-csc^2x

d/dx secx

secxtanx

d/dx cscx

-cscxcotx

if f(x) represents a position function_______ represents the velocity function

f`(x)

if f(x) represents a position function_______ represents the acceleration function

f ``(x)

d/dx f(g(x)

f`(g(x)) * g`(x)

g(x) = inverse of f(x), then g'(x) =

1/f'(g(a))

d/dx [a^x]

(lna)(a^x)

d/dx [logax]

1/(lna)x

d/dx [arcsinx]

1/sqrt(1-x^2)

d/dx [arccosx]

-1/sqrt(1-x^2)

d/dx [arctanx]

1/(1+x^2)

d/dx [arccotx]

-1/(1+x^2)

d/dx [arcsecx]

1/|x|*sqrt(x^2-1)

d/dx [arccscx]

1/|x|*sqrt(x^2-1)