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Finding a Second Derivative Let $f^{\prime \prime}(x)$ be continuous. Show that$\lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}=f^{\prime \prime}(x)$

Using L'Hospital's Rule, we can prove the proposition.

Calculus 1 / AB

Chapter 5

Logarithmic, Exponential, and Other Transcendental Functions

Section 6

Indeterminate Forms and L'Hopita's Rule

Functions

Differentiation

Integrals

Campbell University

Baylor University

University of Michigan - Ann Arbor

Lectures

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All right. So by direct substitution limit is determined to re getting it zero divided by zero, which is an indeterminant form. So therefore we can use low petals rule and we're also going to use the change roles we have to limit as h approaches zero off f crime every sometimes and we have f crime of X plus Age times one minus zero plus f prime off X minus H times negative one all divided by two h So this is equal Thio prime off X plus H minus F crime off X minus H hull divided by to age. Now we tried to wreck substitution again. We're gonna get zero over zero, which is again an indeterminant form. So if we do Moby tell through again, we're gonna get a double prime of X plus H plus f double prime of X minus age, all divided by two. Now we try and direct substitution. You get to find a living we're gonna get This is equal to F double prime off X

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The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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