Calculus Final

continuity definition

let f be defined on an open interval containing a, f is continuous is limit as x approaches a of f(x) equal f(a)

Intermediate value theorem

for a continuous function on a closed interval [a,b] is must take on every value between f(a) and f(b)

product rule

d/dx [f(x) x g(x)] = f(x)*g’(x) + g(x)*f’(x)

quotient rule

d/dx [f(x)/g(x)] = lo*dhi- hi*dlo/(lo²)

constant multiple rule

y=c*f(x) = c * the limit as h approaches 0 of f(x+h)-f(x)/ h

derivative of sin(x)

cos (x)

derivative of cos (x)

-sin (x)

derivative of tan(x)

sec²(x)

derivative of csc(x)

-csc(x)*cot(x)

derivative of sec(x)

sec(x)*tan(x)

derivative of cot(x)

-csc²(x)

the limit as x approaches 0 of sin(x) over x.

(lim_x→0(sinx/x)) equals?

1

the limit as x aproaches 0 of cos(x)-1 over x.

(lim_x→0[cos(x)-1/x]) equals?

0

the limit as x approaches 0 of x over cos(x)-1.

(lim_x→0[x/cos(x)-1]) equals?

DNE

graph of sin^-1(x)

graph of cos^-1(x)

graph of tan^-1(x)

graph of csc^-1(x)