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An $\$ 8,000$ car loan is paid off with $\$ 250$ monthly payments over three years, what monthly interest rate $r$ is the bank charging? It can be shown that this problem translates into the equation $$8000=250\left(\frac{1-(1+i)^{-36}}{i}\right).$$ where $r=12$ i. Find $r .$ Hint: rewrite the above expression as a polynomial, with $x=1+i$A variation of Newton's Method is called the Secant Method. It avoids the use of the derivative. It replaces the derivative by the approximation, $f^{\prime}\left(x_{N}\right) \approx \frac{f\left(x_{N}\right)-f\left(x_{N 1}\right)}{x_{N}-x_{N-1}}$.that is, the derivative is replaced by the slope of the (secant) line through the points $\left(x_{N}, f\left(x_{N}\right)\right)$ and $\left(x_{N-1}, f\left(x_{N-1}\right)\right) .$ Thus, wehave $$x_{N+1}=x_{N}-\frac{f\left(x_{N}\right)\left(x_{N}-x_{N-1}\right)}{f\left(x_{N}\right)-f\left(x_{N-1}\right)} \quad \text { for } N \geq 1.$$ Note that this means we need two starting values near the root, $x_{0}$ and $x_{1}$.

$r=7.56 \%$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 11

Newton s Method

Derivatives

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

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.

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41. I have a wolf. Are you hurt by one minus one? Plus I in the end with negative and power where a equals 18,000 r equals 3 75 and n equals five years Fortune 16 months. So now I'm plugging in my known quantities I have 18,000 equals 3 75 eyes with my ex one minus one plus x with negative way Multiply both sides by the Super Bowl of 3 35 over X. That would give me 48. Yes, equals one minus one over one plus x the city of power multiplying everything by one plus x to the 60 of power I get 48 It's one plus X with the 60th power moves one plus x to this power Buying this one. Bring everything to one side and I have boarding in X finds one plus x to the sixties Power mine It's one plus X with 60 of power plus one equal zero, which is my ex. And now I'm gonna take the derivative of that. No, my prime. I take the product mobile. I'm gonna get 48 finds one plus to the six of negative 60 of power. No positives. You have power Loves 48 x time 60 one plus X to the 59th Power minus six with one less eggs to the 59 hours. So my f fine. It's been a simple morning. A one plus X in the 60th. No. 28 baby one plus X with 59 power minus 60 one plus X to the 59 power. Now I'm gonna factor out right with the factor out at 12. One plus eggs to the 59th Power You can't leave me where Work one plus x plus 2 40 2 40 Thanks, minus five. Distributing that four. So I get a crime equals O on X with 59 hour combined My life terms Wait 2 40 or were X minus one. That is my f frying. Now I get to Newton's equals x So start with 1%. So I get X equals 0.1 except two Quite 008 due to go to except three equals, You know, row seven six, you know to exit four 76 two and exit five equals at six, which so six 286 is 0.76 286%. And that would be our monthly. Never want to figure that out. Compounded her year you were just multiply that fight both. No compounded monthly per year. Compounded monthly.

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