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# Week 11 Problem Set C.
## Reading.
Chapter 4.4 Concavity and curve sketching
Chapter 4.5 Applied optimization
Chapter 4.6 Newton's method
This problem set is on applied Newton's method (4.6).
You are allowed to use a non-graphing scientific calculator for this and on the exam.
## Problems.
### Root finding.
In each of the problem, write down the **iteration formula** we get by using Newton's method, namely the equation $x_{n+1} = \cdots$ in terms of $x_{n}$.
1. Use Newton's method to estimate the solutions of the equation $x^{2} + x - 1 = 0$. Start with $x_{0}=-1$ for the left-hand solution, and with $x_{0}=1$ for the solution on the right (there are two). Then, in each case, what was the iterate $x_{2}$ ?
2. Use Newton's method to estimate the one real solution of $x^{3} + 3x +1 = 0$. Start with the initial guess $x_{0}=0$, find the iterate $x_{2}$, and use your calculator to estimate the real solution to 6 decimal places.
3. Use Newton's method to estimate the two zeros of the function $f(x)=x^{4} + x - 3$. Start with $x_{0} = -1$ for the left-hand zero, and with $x_{0}=1$ for the zero on the right. In each case, write down what was the iterate $x_{2}$. Then, use your calculator to estimate the roots to 6 decimal places.
### Intersection of curves.
Since to find where two curves $y=f(x)$ and $y=g(x)$ intersects is to solve for $x$ such that $f(x)-g(x)=0$, we can turn a intersection of curves problem to a root finding problem, and hence we can apply Newton's method.
In each of the problem, also write down the **iteration formula** we get by using Newton's method, namely the equation $x_{n+1} = \cdots$ in terms of $x_{n}$.
1. The graphs of $y=x^{2}(x+1)$ and $\displaystyle y=\frac{1}{x}$ ($x > 0$) intersects at one point $x=r$ as shown. Use Newton's method to estimate the value of $r$ to 6 decimal places. ![[1 teaching/smc-fall-2023-math-7/week-11/---files/Pasted image 20231110120006.png]]
2. The graphs of $y=\sqrt{x}$ and $y=3-x^{2}$ intersects at one point $x=r$. Use Newton's method to estimate the value of $r$ to four decimal places.
3. The curve $y=\tan(x)$ cross the line $y=2x$ between $x=0$ and $\displaystyle y=\frac{\pi}{2}$. Use Newton's method to find where, to 6 decimal places. Note: Put your calculator in radian mode!
4. At what value(s) of $x$ does $\cos(x)=2x$? Use Newton's method to estimate them to 6 decimal places.
5. At what value(s) of $x$ does $\cos(x)=-x$? Use Newton's method to estimate them to 6 decimal places.
### Reciprocal and root extraction.
1. Given a number $N$, devise an **iteration formula** to estimate $\frac{1}{N}$ using Newton's method. Your iterative formula should look like $x_{n+1} = \cdots$ in terms of $x_{n}$. Hint: You have to come up with a function $f(x)$ first so that its root is $\frac{1}{N}$ to apply Newton's method.
2. Given a number $N$, devise an **iteration formula** to estimate $\sqrt[4]{N}$ using Newton's method. Your iteration formula should look like $x_{n+1} = \cdots$ in terms of $x_{n}$. Hint: You have to come up with a function $f(x)$ first so that its root is $\sqrt[4]{N}$ to apply Newton's method.
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