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AP Calculus AB: 8.2.1 Critical Points

Mathematics5 CardsCreated 8 months ago

This set of flashcards explains how to identify critical points—where the derivative is zero or undefined—and why these points are important in understanding the behavior of functions. It guides through finding critical points by differentiation, solving equations, and considering points of non-differentiability like cusps.

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Critical Points

  • Critical points are the places on a graph where the derivative equals zero or is undefined. Interesting things happen at critical points.

  • To find critical points, take the derivative, set the derivative equal to zero and solve, and find values where the derivative is undefined.

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Key Terms

Term
Definition

Critical Points

  • Critical points are the places on a graph where the derivative equals zero or is undefined. Interesting things happen at critical points.

note

  • Consider this graph of some complicated function. There seem to be several interesting points.

  • At x 1 , x 2 , x 4 , and x 5 ...

Does this function have a critical point at x = 0?

Yes.

Find all of the critical points of p (t).

p (t) = t ^2 + 5t + 6.

t=−5/2

Find all of the critical points of the function f (x) = |3x|

x = 0

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AP Calculus AB: 8.2.1 Critical Points
TermDefinition

Critical Points

  • Critical points are the places on a graph where the derivative equals zero or is undefined. Interesting things happen at critical points.

  • To find critical points, take the derivative, set the derivative equal to zero and solve, and find values where the derivative is undefined.

note

  • Consider this graph of some complicated function. There seem to be several interesting points.

  • At x 1 , x 2 , x 4 , and x 5 , the tangent lines are horizontal, so the derivative of the function is 0 at each of those values.

  • At x 3 , the tangent line is vertical. Its slope is undefined, so the derivative of the function at that point is undefined, too.

  • These points are called critical points. To identify them you need to solve for values that make the derivative equal to zero or undefined.

  • Here is an example of a function that you might want to graph.

  • To find its critical points, first take the derivative.

  • Next, set the derivative equal to zero and solve.

  • Finally, look for places where the derivative is undefined. In this case, the derivative is defined for all real numbers.

  • You will need to use the power rule to take the derivative of this function.

  • First, notice that there are no values of x that make the derivative equal to zero.

  • Next, look for values of x for which the derivative is
    undefined. One way to do this is to set the denominator equal to zero and solve. In this case, the derivative is undefined for x = 0, even though the function itself is defined at that point. Therefore, you may have a cusp point at that x-value.

Does this function have a critical point at x = 0?

Yes.

Find all of the critical points of p (t).

p (t) = t ^2 + 5t + 6.

t=−5/2

Find all of the critical points of the function f (x) = |3x|

x = 0