Binary and Decimal Conversions

a$)$ Convert the following decimal numbers to binary:

$)12\mathrm{.}375$

$)0\mathrm{.}021$

b$)$ Convert the following binary numbers to decimal:

$)1011\mathrm{.}101$

$)11001\mathrm{.}011$

Analytic, Asymptotic, and Numerical Approaches

a$)$ Compare analytic, asymptotic, and numerical approaches. Explain their similarities, differences, and the importance of each in engineering applications.

b$)$ Solve the quadratic equation $2x^2-4x+1=0$ using an analytic approach $($quadratic formula$),$ then discuss how this might be handled using a numerical method $($like Newton's method$)$ for an approximate solution.

Well$-$Posedness

Explain the concept of well$-$posedness in mathematical modeling. What are the three criteria that a problem must satisfy to be considered well$-$posed?

$4.$ Significant Digits

a$)$ Define significant digits and explain their importance in numerical calculations. Provide examples where the use of significant digits impacts the accuracy of results.

b$)$ Perform the following calculation using four significant digits:

$3\mathrm{.}456572\mathrm{.}12312$

Compare this with the result using full precision. Discuss the effect of significant digits on the accuracy of the result.

$5.$ Underflow and Overflow

a$)$ Define underflow and overflow in the context of floating$-$point arithmetic. Provide examples where these issues might arise in engineering computations and explain how they can affect the results.

b$)$ Consider the operation

$1\mathrm{.}0{10}^{-310}1\mathrm{.}0{10}^{300}$

Perform this computation in single precision and double precision and discuss the results in the context of underflow or overflow.

$6.$ Machine Epsilon

a$)$ Define machine epsilon and explain its significance in numerical methods. How does machine epsilon impact the accuracy of floating$-$point operations?