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Show that $a_{\mathrm{k}}=\frac{f^{(k)}(0)}{k !}$ where the symbol $$k !=k(k-1)(k-2) \cdots 2 \cdot 1 . \text { Thus, } 4 !=4(3)(2)(1)=24 . \text { The symbol }$$ 0! is defined to be equal to 1 . The symbol " $"$ is read factorial.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

00:34

Show that $i^{4 k}=1,$ for…

02:53

Show that $\sum_{k=1}^{\in…

01:17

Show that $i^{4 k-1}=-i,$ …

04:09

04:05

Let $n !=n(n-1)(n-2) \cdot…

02:21

Show that $\left(\begin{ar…

01:28

Find a value of the consta…

01:24

$$\text { Show that } …

The following problem we want to show that a sub k um is equal to, yeah, F cash shape prime of zero over K factorial. Mhm. Okay. So um we know that depending on our function, if we have a f k um depending on the derivative that we're taking, if it's the first derivative, Then we have a prime of zero Um over one factorial, which is one. Or we could have a not which would be just f of zero over Our zero factorial, which is equal to one.

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