Calculus Midterm Formula and Theorem Review

Flashcards with special limits and derivatives, plus locations of horizontal asymptotes, the Squeeze Theorem and L'Hospital's Rule.

lim as x → 0 of sin(x)/x =

1

lim as x → 0 of (1-cos(x))/x =

0

Squeeze Theorem

If g(x)≤f(x)≤h(x) and lim as x→a g(x)=L and lim as x→a h(x)=L, then lim as x→a f(x)=L

If the denominator grows faster the horizontal asymptote…

is at 0

If the numerator grows faster the horizontal asymptote…

does not exist

If the denominator and numerator grow at the same rate, the horizontal asymptote…

is the leading coefficient in numerator divided by leading coefficient in denominator

Derivative of a^x

a^x*ln(a)

Derivative of log_ax

ln(a)*1/x

Derivative of sin(x)

cos(x)

Derivative of cos(x)

-sin(x)

Derivative of tan(x)

sec²(x)

Derivative of cot(x)

-csc²(x)

Derivative of sec(x)

sec(x)*tan(x)

Derivative of csc(x)

-csc(x)*cot(x)

Derivative of f(g(x))

f’(g(x))*g’(x)

Derivative of f^-1(x)

1/f’(f^-1(x))

Derivative of arcsin(x)

1/sqrt(1-x²)

Derivative of arccos(x)

-1/sqrt(1-x²)

Derivative of arcsec(x)

1/(|x|*sqrt(x²-1))

Derivative of arccsc(x)

-1/(|x|*sqrt(x²-1))

Derivative of arctan(x)

1/(x²+1)

Derivative of arccot(x)

-1/(x²+1)

L’Hospital’s Rule

If f(a)=0 and g(a)=0 and the lim as x approaches a of f(x)/g(x) equals 0/0 or infinity/infinity, then the lim as x approaches a of f(x)/g(x) equals f’(a)/g’(a)