Definitions/Theorems to Know

Question 1

Precise Definition of a Limit

Answer 1

Let 𝑓(𝑥) be defined for all 𝑥 ≠ 𝑎 over an open interval containing a. Let L be a real number. Then,

lim 𝑥 → 𝑎 𝑓(𝑥) = 𝐿

If, for every 𝜀 > 0, there exists a 𝛿 > 0, such that if 0 < |𝑥 − 𝑎| < 𝛿, then |𝑓(𝑥) − 𝐿| < 𝜀.

Question 2

Proof Statement for Precise Definition of a Limit

Answer 2

Choose a 𝛿 = ___. Thus, it follows that if 0 < |𝑥 − 𝑎| < ___, then |𝑓(𝑥) − 𝐿| < 𝜀. This completes the proof.

Question 3

A function is continuous if…

Answer 3

It can be drawn on a graph without lifting the pencil— meaning there’s no sudden jumps, holes, or breaks—and at every point the function’s value is equal to the limit at that point.

Question 4

A function is differentiable if…

Answer 4

Its derivative exists at every point in its domain— meaning it is continuous and doesn’t have any sudden changes in slope.

Question 5

Conditions for Mean Value Theorem and Conclusion

Answer 5

If…
1) Function is continuous on the closed interval [a,b]
2) The function is differentiable on the open interval (a,b)
Then…
There exists a point c in the interval (a,b) such that f’(c) is equal to the function’s average rate of change over [a,b], f(b) - f(a) / b - a

Question 6

Steps to Solving Can Minimization Problem

Answer 6

1) Form surface area equation
2) Express h in terms of r using volume equation and plug into area equation, this is now the function
3) Determine domain of the function in context of the problem (0,∞)
4) Take the derivative of the function and find the critical points
5) Use the first derivative test to confirm the minimum
6) Find the full dimensions