Problem 1
T/F: Given a function \(f(x)\), Newton's Method produces an exact solution to \(f(x)=0\)
Problem 1
T/F: Implicit differentiation is often used when solving "related rates" type problems.
Problem 1
T/F: An "optimization problem" is essentially an "extreme values" problem in a "story problem" setting.
Problem 1
\(\mathrm{T} / \mathrm{F}:\) Given a differentiable function \(y=f(x),\) we are generally free to choose a value for \(d x,\) which then determines the value of \(d y\).
Problem 2
\(\mathrm{T} / \mathrm{F}:\) In order to get a solution to \(f(x)=0\) accurate to \(d\) places after the decimal, at least \(d+1\) iterations of Newtons' Method must be used.
Problem 3
Water flows onto a flat surface at a rate of \(5 \mathrm{~cm}^{3} / \mathrm{s}\) forming a circular puddle \(10 \mathrm{~mm}\) deep. How fast is the radius growing when the radius is: (a) \(1 \mathrm{~cm}\) ? (b) \(10 \mathrm{~cm} ?\) (c) \(100 \mathrm{~cm} ?\)
Problem 3
In Exercises 3-8, the roots of \(f(x)\) are known or are easily found. Use 5 iterations of Newton's Method with the given initial approximation to approximate the root. Compare it to the known value of the root. $$ f(x)=\cos x, x_{0}=1.5 $$
Problem 3
Find the maximum product of two numbers (not necessarily integers) that have a sum of 100 .
Problem 4
A circular balloon is inflated with air flowing at a rate of \(10 \mathrm{~cm}^{3} / \mathrm{s}\). How fast is the radius of the balloon increasing when the radius is: (a) \(1 \mathrm{~cm} ?\) (b) \(10 \mathrm{~cm}\) ? (c) \(100 \mathrm{~cm} ?\)
Problem 4
The roots of \(f(x)\) are known or are easily found. Use 5 iterations of Newton's Method with the given initial approximation to approximate the root. Compare it to the known value of the root. $$ f(x)=\sin x, x_{0}=1 $$
