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Magazine Problem 1
Convert the following base- 2 numbers to base- 10 : \((a) 101101,\) (b) \(101.101,\) and \((\mathrm{c}) 0.01101\).
Problem 5
Evaluate \(e^{-5}\) using two approaches \\[e^{-x}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{3 !}+\cdots\\] and \\[e^{-x}=\frac{1}{e^{x}}=\frac{1}{1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3 !}+\cdots}\\] and compare with the true value of \(6.737947 \times 10^{-3}\). Use 20 terms to evaluate each series and compute true and approximate relative errors as terms are added.
Problem 7
(a) Evaluate the polynomial \\[y=x^{3}-7 x^{2}+8 x-0.35\\] at \(x=1.37 .\) Use 3 -digit arithmetic with chopping. Evaluate the percent relative error. (b) Repeat (a) but express \(y\) as \\[y=((x-7) \cdot x+8) x-0.35\\] Evaluate the error and compare with part (a).
Problem 8
Calculate the random access memory (RAM) in megabytes necessary to store a multidimensional array that is \(20 \times 40 \times 120\) This array is double precision, and each value requires a 64 -bit word. Recall that a 64 -bit word \(=8\) bytes and 1 kilobyte \(=2^{10}\) bytes. Assume that the index starts at 1.
Problem 9
Determine the number of terms necessary to approximate \(\cos x\) to 8 significant figures using the Maclaurin series approximation \\[ \cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !}+\frac{x^{8}}{8 !}-\cdots \\] Calculate the approximation using a value of \(x=0.3 \pi .\) Write a program to determine your result.
Problem 11
How can the machine epsilon be employed to formulate a stopping criterion \(\varepsilon_{s}\) for your programs? Provide an example.
