Limits
The limit as x approaches 3 of (x2-9)/(x2-4x+3).
What is 3?
The limit as h approaches 0 of (f(a+h)-f(a))/h.
What is the limit definition of the derivative?
d/dx [3x+6]
What is 3?
The line tangent to f(x)=3x2 at x=-2.
What is y-6=-12(x+2)?
The critical point(s) of f(x)=x3-9x2+24x+67.
What is x=2,4?
The limit as x approaches infinity of 1/x3.
What is 0?
This definition requires that f(a) exists, the limit as x approaches a of f(x) exists, and that those two values are equal to each other.
What is Continuity?
d/dx [sin(x)cos(x)]
What is cos2x - sin2x?
The line secant to f(x)=cosx over x=[0,pi].
What is y-1=(-2/pi)(x-0)?
The inflection points of f(x)=xex.
What is x=-2?
The limit as x approaches 0 of 1/x.
What is "the limit does not exist"?
This theorem states that if f is a continuous function over (a,b), and there is some value K such that f(a)<K<f(b). Then there exists c in (a,b) such that f(c)=K.
What is the Intermediate Value Theorem?
d/dx [(cosx)1/7]
What is (-1/7)(cosx)-6/7(sinx)?
The line normal to f(x)=3x3-4x2+3x+1 at x=0.
What is y-1=(-1/3)(x-0)?
The maximum of f(x)=(x-2)(x-3)(x-4).
What is infinity?
The limit as x approaches negative infinity of (3x5+1)/(2x4-6x2+3).
What is negative infinity?
This theorem states that for functions where g(x)<f(x)<h(x), and the limits of g(x) and h(x) are equal, then the limit of f(x) is that same value.
What is the Squeeze Theorem?
d/dx [x/sqrt(1+x2)]
What is (1-x2)/(1+x2)3/2?
The line tangent to x2+y2=1 at x=1/2.
What is y-(sqrt(3)/2)=(-sqrt(3)/3)(x-(1/2))?
The interval of when f(x)=3x3+x2 is concave up.
What is x>(-1/9)?
The limit as x approaches 0 of sin(x)/x.
What is 1?
This theorem states that for some function f that is continuous over [a,b] and differentiable over (a,b), there exists some c such that f'(c)=(f(b)-f(a))/(b-a).
What is the Mean Value Theorem?
d/dx [y=43x+1]
What is (3ln4)43x+1?
The line tangent to 3xy+y3-6x=4 at (-1,1).
What is y=-1?
The dimensions of a rectangle that minimizes perimeter with respect to a diagonal of 5.
What is a length and width of sqrt(5)?