Unit 5 Flashcards
How can you tell if a slope is increasing or decreasing based on tangent lines?
If the tangent lines slopes (f’(x)) are positive, the line is increasing
What are the two scenarios in which critical points occur?
When f’=0
When f’ does not exist (denominator=0)
What is the first step in using the first derivative test?
FIND DOMAIN
How do we find a relative maximum or minimum?
First find domain, then run first derivative test to find the critical numbers, then plug the derivative into a sign analysis to see if the critical points are max or min
What is the notation for the second derivative?
d^2y/dx^2
What does the second derivative test show about a graph? Given f’(x)=0 at this point
If f’(x)=0 and f’’(x) is positive, the point is a relative minimum and the graph is concave up.
If f’(x)=0 and f’’(x) is negative, the point is a relative maximum and the graph in concave down
What happens if f’(x) is 0 and f’’(x) is 0?
The test fails and we have no idea if it’s relative max min or an inflection point
How do we find inflection points?
Make f’’(x) equal to 0
Make all x’s 0
Then put the second derivative on a sign analysis to see if it inflected at that point
What are the steps to curve sketching?
- FIND DOMAIN
- Find the x and y intercepts
- Find the asymptotes if any
- Find the first derivative and make sign analysis to see relative max min
- Make second derivative and make sign analysis to see any inflection points and concave up or down
How do you find the horizontal asymptote?
If the x in the numerator is greater than in the denominator, you divide the denominator into the numerator and the result is the horizontal asymptote
If the x in the numerator is equal to the x in the denominator, you divide the numerator leading coefficient by the denominator coefficient and that’s the asymptote
If the x in the numerator is less than the x in the denominator, y=0
