Continuous functions, discontinuous functions, rules of differentiation.

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timer Asked: Jun 3rd, 2018
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Question description

1. Prove (using the defi nition of continuity), that f(x) = x^2 - 4 is a continuous function.
2. Give an example of a discontinuous function and explain why it is discontinuous.
3. Find the average rate of change of f(x) = x^3 from x1 = 1 to x2 = 3.
4. Find the derivative of f(x) = x^2 at the point x0 = 2 using the de nition of the derivative.
5. Using the rules of di erentiation, find the derivative of f(x) = 5x^4 + 3x.
6. Using the rules of di erentiation, fi nd the derivative of f(x) = x^3 e^x^2
7. Using either the First or Second Derivative Test, nd all maxima and minima of f(x) = x^3 + 2x^2 + 1.

Tutor Answer

Borys S
School: UCLA

The solutions are ready, please ask if something is unclear.

1. 𝑓(π‘₯) = π‘₯ 2 βˆ’ 4 is a continuous function for any π‘₯.
Proof. Consider |𝑓(π‘₯) βˆ’ 𝑓(π‘₯1 )| = |π‘₯ 2 βˆ’ π‘₯12 | = |π‘₯ βˆ’ π‘₯1 ||π‘₯ + π‘₯1 | =
= |π‘₯ βˆ’ π‘₯1 |(|(π‘₯1 βˆ’ π‘₯) + 2π‘₯|) ≀ |π‘₯ βˆ’ π‘₯1 |(|π‘₯1 βˆ’ π‘₯| + 2|π‘₯|).
We need to choose 𝛿(πœ€, π‘₯) > 0 for any given π‘₯ and πœ€ > 0 such that |π‘₯ βˆ’ π‘₯1 | < 𝛿(πœ€, π‘₯)
guarantees that |𝑓(π‘₯) βˆ’ 𝑓(π‘₯1 )| < πœ€.
Suppose |π‘₯| < 𝑀 (such 𝑀 exists for any π‘₯). We can always choose 𝛿 < 1.
πœ€

2𝑀+1

This way if we choose 𝛿(πœ€, π‘₯) = min (1, 2𝑀+2) we’ll get |𝑓(π‘₯) βˆ’ 𝑓(π‘₯1 )| ≀ πœ€ 2𝑀+2 < πœ€, ∎

0, π‘₯ ≀ 0
2. Consider a function 𝑓(π‘₯) = {
.
1, π‘₯ >...

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Anonymous
Goes above and beyond expectations !

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