Quiz:Product rule for higher derivatives: Difference between revisions

Latest revision as of 06:06, 11 April 2024

ORIGINAL FULL PAGE: Product rule for higher derivatives
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ALSO CHECK OUT: Quiz (multiple choice questions to test your understanding) |

For a quiz that tests all the differentiation rules together, see Quiz:Differentiation rules.

Practical

Suppose $f$ and $g$ are continuous functions at $x_{0}$ . Suppose $f(x_{0})=f'(x_{0})=f''(x_{0})=f'''(x_{0})=g(x_{0})=g'(x_{0})=g''(x_{0})=g'''(x_{0})=1$ . What is $(f\cdot g)'''(x_{0})$ ?

Significance

Qualitative and existential significance

Suppose $f$ and $g$ are continuous functions at $x_{0}$ . Suppose we know that $f$ is 5 times differentiable at $x_{0}$ and $g$ is 3 times differentiable at $x_{0}$ . What is the maximum number of times we can be sure (from this information) that $f\cdot g$ is differentiable at $x_{0}$ ?

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 ==Practical==
+<quiz display=simple>
+{Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose <math>f(x_0) = f'(x_0) = f''(x_0) = f'''(x_0) = g(x_0) = g'(x_0) = g''(x_0) = g'''(x_0) = 1</math>. What is <math>(f \cdot g)'''(x_0)</math>?
+|type="()"}
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+</quiz>
+==Significance==
+===Qualitative and existential significance===
 <quiz display=simple>
 {Suppose <math>f</math> and <math>g</math> are continuous functions at <math>x_0</math>. Suppose we know that <math>f</math> is 5 times differentiable at <math>x_0</math> and <math>g</math> is 3 times differentiable at <math>x_0</math>. What is the maximum number of times we can be sure (from this information) that <math>f \cdot g</math> is differentiable at <math>x_0</math>?