Theorems
Where is this function continuous?
x^7-6x^5+4x-12
Everywhere. Remember that polynomials are continuous for all real numbers.
What is the horizontal asymptote if the degree of the numerator is greater than the degree of the denominator.
There is none.
Find the derivative of the following function using the definition.
f(x)= (1-x^2)/3
f'(x) = -2/3x
What is the 13th derivative on sinx?
cosx
Where is the function continuous?
(x^2+4x-21)/(x^2-x-6)
(-oo, -2)uu(-2,3)uu(3, oo)
Remember that rational functions are continuous on their domains (everywhere they are defined).
What is the horizontal asymptote if the degrees of the numerator and denominator are equal.
Ratio of the leading coefficients
Find the derivative using the definition.
sqrt(2-6x)
f'(x)= (-3)/sqrt(2-6x)
What is the 45th derivative of cosx?
-sinx
Where is the function continuous?
f(x) = tanx
Everywhere except odd multiples of pi/2.
Remember that root and trig functions are continuous everywhere on their domains.
What is the horizontal asymptote if the degree of the numerator is less than the degree of the denominator.
y=0
Find the derivative
7x^2+12x
14x+12
What is the tangent line to y=x^4 at x=2?
y=32x-48
Prove that the following function has a real root between x=6 and x=7.
x^3-7x^2+4
Intermediate Value Theorem
Polynomial so continuous.
a=6, b=7 f(6)=-32<0, f(7)=4>0
0 is between f(a) and f(b), so c must be between a and b.
This is where vertical asymptotes CAN be found.
Unremovable zeroes of the denominator.
Find the derivative
(3sqrtx^7)/(x^(4/5)sqrtx)
(33/5)x^(6/5)
What is the normal line to y=x^4 at x=2
y=-1/32x+257/256
Find the
lim_(x->0)(cos(1/3x)(x^5+2x^3))
Use the Sandwich/Squeeze Theorem.
What are all the asymptotes of
(5x^3+7x)/(2x^3-6x^2)
y= 5/2, x=0, x=3
What is the derivative
4cosx- x^6/13
-4sinx-6/13x^5
Where are the maximums or minimums of
f(x)=4sinx
Where -4cosx=0
Odd multiples of pi/2