House of Calculus
Finish this theorem
A limit exists if...?
limit as x approaches c from the left of f(x) exists
limit as x approaches c from the right of f(x) exists
limit as x approaches c from the left of f(x) equals the limit as x approaches c from the right of f(x)
State the Intermediate Value Theorem
If f(x) is continuous on [a,b]
then there exists a c on (a,b)
such that f(c) is between f(a) and f(b)
f(x) = (cos(3x))2, find f ' (x)
f ' (x) = -6cos(3x)sin(3x)
Given f (x) = x3, what are the absolute extrema on [-2,5]?
Abs max is 125
Abs min is -8
Limit as h approaches 0 of (f(x+h)-f(x))/(h).
Finish this theorem
A function is continuous at x=a if...
Limit as x approaches a of f(x) exists
f(a) exists
Limit as x approaches a of f(x) = f(a)
State the Extreme Value Theorem
If f(x) is continuous on [a,b]
then there exists at least one Abs Max and at least one Abs Min
Find dy/dx if 3x2y + 2x = y3 - 5.
dy/dx = (-6xy - 2) / (3x2 - 3y2)
Find the point of inflections for f(x) when
f " (x) = x2(x-1)(x+3).
-3, 1
Finish this statement...
Differentiablility applies
Continuity
Finish this theorem
A function is differentiable at x=c if...
f(x) is continuous
Limit as x approaches c from the left of f ' (x) exists
Limit as x approaches c from the right of f ' (x) exists
Limit as x approaches c from the left of f ' (x) equals Limit as x approaches c from the right of f ' (x)
State the Mean Value Theorem
If f(x) is continuous on [a,b] and differentiable on (a,b)
then there exists a c on (a,b)
such that f ' (c) = (f(b)-(f(a))/(b-a)
State the derivatives of the 6 trig functions
d/dx (cos x) = -sin x
d/dx (sin x) = cos x
d/dx (tan x) = sec2 x
d/dx (csc x) = -csc x cot x
d/dx (sec x) = sec x tan x
d/dx (cot x) = -csc2 x
Given f (x) = (x-1)3, where is f(x) increasing? Explain reasoning.
Write the equation of the normal line at the point
(pi/4,1)
if
g(k) = tan k
L(x) = 1 -1/[sec^2 (pi/4)] (x-pi/4)
simplifies to
L(x) = 1-2(x-pi/4)
Draw and explain a graph that is continuous but not differentiable.
Answer will vary
Does the mean value guarantee a c such that
f ' (c) = 1 for f(x) = 1/x on the interval from [-2,2]?
No, f(x) = 1/x is not continuous on [-2,2].
Evaluate limit as x approaches 5 of (x2 - 25)/(x-5).
10
Given f (x) = (x-1)3, where is f(x) concave up? Explain reasoning.
(1,inf)
What is the equation of the horizontal asymptote of m(n) = (n-1)/(n+3)?
y = 1
Given f(x) = |x|, describe the intervals that f(x) is differentiable.
(-inf,0) and (0,inf)
State Rolle's Theorem
If f(x) is continuous on [a,b] and differentiable on (a,b) and f(a) = f(b)
then there exists a c on (a,b)
such that f ' (c) = 0
What is the equation of the tangent line of
g(x) = x2 - 1 at x = 3?
L(x) = 8 + 6 (x-3)
If it is given that f " (3) = -3 and f " (4) = 5, what additional piece of information do you need so f (x) must have a point of inflection on the interval [3,4]?
f(x) is continuous on the [3,4]
Evaluate d/dx (sin-1 x)
1/sqrt(1-x2)