Unit 3: Differentiation: Composite, Implicit, and Inverse Vocab

Are you excited to learn Unit 3?

• Yes, I am excited to learn Unit 3.
• I'm indifferent about Unit 3.
• I already know everything about Unit 3.
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What is true about Rates of Change and derivatives?

• Rates of change and derivatives are fundamentally related; derivatives quantify the rate of change of a function.
• Rates of change are only applicable to linear functions.
• Derivatives can only be calculated for constant functions.
• Derivatives measure the total area under a curve.

We say that x=c is a critical point of the function f(x) if f(c) exists and if either of the following are true.

What does this demonstrate?

• It demonstrates the conditions under which critical points are identified in calculus.
• It defines the limits of the function's domain.
• It indicates the symmetry of the function.
• It shows the maximum values of the function.

If (x)≤f(c) for every x in the domain we are working on.

We say....

• 1. 
     We say that f(x) has a relative (or local) maximum at x=c

• 1. We say that f(x) has a relative (or local) minimum at x=c if (x)≥f(c) for every x in some open interval around x=c

• We say that f(x) has an absolute (or global) maximum at x=c

• We say that f(x) has an absolute (or global) minimum at x=c

1. f(x)≤f(c) for every x in some open interval around x=c
   

We say...

• 1. 
     We say that f(x) has a relative (or local) maximum at x=c

• 1. We say that f(x) has a relative (or local) minimum at x=c if (x)≥f(c) for every x in some open interval around x=c

• We say that f(x) has an absolute (or global) maximum at x=c

• We say that f(x) has an absolute (or global) minimum at x=c

1. f(x)≥f(c) for every x in the domain we are working on.
   

We say...

• 1. 
     We say that f(x) has a relative (or local) maximum at x=c

• 1. We say that f(x) has a relative (or local) minimum at x=c if (x)≥f(c) for every x in some open interval around x=c

• We say that f(x) has an absolute (or global) maximum at x=c

• We say that f(x) has an absolute (or global) minimum at x=c

f(x)≥f(c) for every x in some open interval around x=c

We say...

• 1. 
     We say that f(x) has a relative (or local) maximum at x=c

• 1. We say that f(x) has a relative (or local) minimum at x=c

• We say that f(x) has an absolute (or global) maximum at x=c

• We say that f(x) has an absolute (or global) minimum at x=c

What is an extrema?

• Extrema are the points where a function is continuous.
• Extrema refer to the range of a function.
• Extrema are the average values of a function.
• Extrema are the maximum or minimum values of a function.

Suppose that f(x)f(x) is continuous on the interval [a,b][a,b] then there are two numbers a≤c,d≤ba≤c,d≤b so that f(c)f(c) is an absolute maximum for the function and f(d)f(d) is an absolute minimum for the function.

• There exist numbers c and d in [a,b] such that f(c) is an absolute maximum and f(d) is an absolute minimum.
• f(x) is not continuous on the interval [a,b].
• f(c) is always less than f(d) for any c and d in [a,b].
• There are no absolute maximum or minimum values in the interval.

Extreme Value Theorem - How does this relate with derivatives?

• The Extreme Value Theorem states that derivatives are always zero at extrema.
• Derivatives are not relevant to the Extreme Value Theorem.
• The Extreme Value Theorem only applies to linear functions.
• The Extreme Value Theorem relates to derivatives by identifying critical points where local extrema may occur.