Derivative Fails to Exist

cusp, sharp turn discontinuity vertical tangent line

smooth curve continuous horizontal tangent line

cusp, sharp turn discontinuity horizontal tangent line

smooth curve continuous vertical tangent line

$$f$$ has an inflection point

$$f$$ has a relative max or min

What does the Extreme Value Theorem remind us to do?

Check the endpoints! They might be the max/min.

Check the inflection points! They might be the max/min.

Check the endpoints! They give the secant slope.

Check the inflection points! They are when the derivative is 0.

Using the nth term test If $$\lim_{n\rightarrow\infty}a_n=0$$ then the series

the test is inconclusive - pick a different one

Using the nth term test If $$\lim_{n\rightarrow\infty}a_n\ne0$$ then the series

the test is inconclusive - pick a different one

A p-series where $$p>1$$ then the series will...

the test is inconclusive - pick a different one

In a geometric series, if $$\left|r\right|\ge1$$ then the series will...

test is inconclusive, pick a different one

In a geometric series, if $$\left|r\right|

test is inconclusive, pick a different one

If $$a_n=f\left(n\right)$$ is continuous, positive, and decreasing, and $$\int_1^{\infty}f\left(n\right)dn$$ converges then $$\sum_{n=1}^{\infty}a_n$$ will...

test is inconclusive - pick a different one!

If $$a_n=f\left(n\right)$$ is continuous, positive, and decreasing, and $$\int_1^{\infty}f\left(n\right)dn=\pm\infty$$ then $$\sum_{n=1}^{\infty}a_n$$ will...

test is inconclusive - pick a different one!

In the limit comparison test, if $$\lim_{n\rightarrow\infty}\left(\frac{a_n}{b_n}\right)=a\ finite,\ positive\ number$$ then....

$$a_n$$ and $$b_n$$ either both converge or both diverge

$$a_n$$ and $$b_n$$ both diverge

$$a_n$$ and $$b_n$$ both converge

the test is inconclusive - pick a different oen!

Based on the Ratio Test for Convergence, if $$\sum_{n=1}^{\infty}a_n$$ has positive terms and $$\lim_{n\rightarrow\infty}\left(\frac{a_{\left(n+1\right)}}{a_n}\right)

test is inconclusive - pick a different one!

Based on the Ratio Test for Convergence, if $$\sum_{n=1}^{\infty}a_n$$ has positive terms and $$\lim_{n\rightarrow\infty}\left(\frac{a_{\left(n+1\right)}}{a_n}\right)>1$$ then the series...

test is inconclusive - pick a different one!

Based on the Ratio Test for Convergence, if $$\sum_{n=1}^{\infty}a_n$$ has positive terms and $$\lim_{n\rightarrow\infty}\left(\frac{a_{\left(n+1\right)}}{a_n}\right)=1$$ then the series...

test is inconclusive - pick a different one!

BC Calculus Memorization

Authored by Chloe Dale

Mathematics

12th Grade

Trig Value covered

Used 2+ times

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3.142

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