Limits
Definitions, Theorems
Derivatives part 1
Derivatives part 2
Implicit Differentiation
100
(4x^2-3x+2)/(3x^2-1) as x--> infinity
4/3
100
Let f be a continuous function on a closed interval [a,b]. If f(a)<0 and f(b)>0, what theorem guarantees the existence of a root of f?
The intermediate value theorem IVT!
100
Find the derivative of sin x.
cos x
100
d/dx (x^3-x)
3x^2-1
100
Find y' if x^2+y^2=17
-x/y
200
(x^6-1)/(x^3-1) as x-->1
2
200
What does it mean for a function f to be continuous at x=a?
f(x) approaches f(a) as x approaches a
200
Find the derivative of e^(2x^2).
4xe^(2x)
200
d/dx (cos x - x)
-sin x - 1
200
Find y' if x^(1/2)+y^(1/2)=9. Your answer should not contain exponents. Instead, use a sqrt.
-sqrt(y/x)
300
3(1-cos(x))/x^2 as x-->0
3/2
300
Give an example of a function which is not differentiable at x=2.
...
300
Find the derivative of arctan x.
1/(1+x^2)
300
d/dx(sin(ln(x)))
cos(ln(x))/x
300
Find sqrt(y') if x^3-y^3=4
x/y
400
(2^x-1)/x as x-->0
ln(2)
400
Fill in the blanks: If a function is ________ at x=a, then it is __________ at x=a.
differentiable, continuous
400
Find the derivative of (tan x)(sec x).
sec^3 x + tan^2 x sec x
400
d/dx (sqrt(cos(x)))
-sin x / (2 sqrt(cos(x)))
400
Find y' if 1=sin(xy)
-ycos(xy)/x
500
ln(1+x)/x as x-->0
1
500
What does the squeeze theorem state?
If f(x)
500
Find the derivative of |x|. Your answer should not be a piecewise function.
|x|/x for x != 0.
500
d/dx (sin(e^(x sin x)))
cos(e^(x sin x)) e^(x sin x) (sin x + x cos x)
500
Find y' if y=x^x. Hint: take ln of both sides.
(1+ln x)x^x
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