Theorems in Calculus
What condition(s) must be meant for a function in Mean Value Theorem?
a. The function is continuous
b. The function is either strictly decreasing or increasing and is continuous.
c. The function is differentiable and continuous
d. The function is differentiable
What is the difference between a local and absolute(global) maximum of a function?
Local maximum- is the largest value of a function within a given range of that function
Absolute(global) - is the largest value of the entire function.
DOUBLE JEOPARDY(double points will be awarded if you not only select the correct answer(s), but then explain why.
Select all the instances when a limit does not exist:
a. jump discontinuity
b. removable discontinuity
c. vertical asymptote
d. corners on a graph
A and C
Jump discontinuity - the left hand limit and right hand limit exist, but do not approach the same value
vertical asymptote - There is unbounded increase or decrease(infinite limit)
State the three conditions that must be met in Rolle's Theorem and state what you can conclude if these conditions are met.
1. [a,b] must be continuous
2. (a,b) differentiable
3. f(a) = f(b)
Conclude:
The derivative of some point c between a and b is equal to 0
If the function f(x) has an absolute minimum at c, what can be said about the derivative of f(c)?
a. The derivative of f(c) is positive
b. The derivative of f(c) is negative
c. The derivative of f(c) = 0
d. Need more information
Correct Answer: D
In order for a function to have an absolute minimum, the point must have a derivative of either 0 or does not exist. The derivative at c could possibly be 0, but we need to also know that could also be does not exist. An example of this would be the function |x|.
True or False:
If the derivative of c exists, then the limit as x approaches c of f(x)= f(c)
True, if the derivative of a function exists, then it is continuous which means the right limit and left limit must equal each other.
True or False:
If f(2) < 0 and f(5) > 0, then there exists a number c between 2 and 5 such that f(c) = 0.
False, it does not state whether the function is continuous. In IVT, the function must be continuous, and in this case it is not known.
True or False
If the second derivative of f(4) = 0, then the point (4,f(4)) is a point of inflection
False
When the second derivative at a point is 0, this means this point could possibly be a point of inflection, but you need more information such as if the second derivative changes signs at this point to determine if it is a point of inflection.
Write out the formula to find the first derivative
lim h→0 [f(x+h) - f(x)]/h