Flashcards in 8.2.1 Critical Points Deck (5):
• 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.
- 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?
Find all of the critical points of p (t).
p (t) = t ^2 + 5t + 6.