Calc Graph Analysis

Finding Extrema

1. Find Endpoints By Plugging in Interval

2. Find Derivative

3. Set Derivative = 0

4. Plug x Into f(x)

5. Compare possible values.

Rolles Theorem

1. Check Endpoints

2. Find Derivative

3. Set Derivative = 0

4. Check Interval

Mean Value Theorem

1. Find f(a) & f(b)

2. AROC

3. Find Derivative

4. Set Derivative = AROC

First Derivative Test

1. Find Derivative

2. Critical Points

3. Test Intervals

4. Classify

Second Derivative Test

1. Find First Derivative

2. Find Second Derivative

3. Set Derivative = 0

4. Test Interval

5. Classify

Extreme Value Theorem

If a function f(x) is continuous on a closed, bounded interval [a,b], then f(x) must attain an absolute maximum f(c) & and absolute minimum f(d) for some c in [a,b].

1. Verify Conditions

2. Find Critical Points

3. Filter Points

4. Evaluate Endpoints and Points

5. Identify Extrema

Intermediate Volume Theorem

If a function is continuous on a closed interval [a,b] & y is any value between f(a) or f(b), then there exists at least one # c in the open interval (a,b) such that f(c) = y.

1. Verify Continuity

2. Evaluate Endpoints

3. Confirm Target Value

4. Conclude Existence