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Quantitative Finance

Numerical Methods

Concept Quizzes

Root Approximation - Bisection
Newton Raphson Method

Root Approximation - Bisection

The equation $\displaystyle e^x = \frac{1}{x}$ has a solution somewhere between $0$ and $1.$ Equivalently, we seek a root of the continuous function $x e^{x}-1$ in the interval $(0,1).$ If we apply the bisecton method 4 times, which of the following intervals will we end up with?

\left[\frac{3}{4}, \frac{5}{8}\right]

\left[\frac{1}{2}, \frac{9}{16}\right]

\left[\frac{9}{16}, \frac{5}{8}\right]

\left[\frac{1}{2}, \frac{5}{8}\right]

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by Brilliant Staff

One of the roots of the equation $\displaystyle f(x)=3x+\sin x-e^x = 0$ lies between $0$ and $0.5.$ If we apply the bisecton method 5 times, which of the following intervals will we end up with?

\left[0.34375, 0.359375\right]

\left[0.3125, 0.328125\right]

\left[0.328125, 0.34375\right]

\left[0.359375, 0.375\right]

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by Brilliant Staff

Determine the value of $\displaystyle \sqrt[3]{5}$ by using the bisecton method. Let the width of the final interval less than $0.02,$ and start with the interval $[1,2].$ Then, guess the upper boundary of the final interval as the value of $\displaystyle \sqrt[3]{5}.$

1.688 1.719 1.714 1.706

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by Brilliant Staff

The equation $\displaystyle (x-2)^3+(x-2)^2-1=0$ has a root between $2$ and $3.$ If we apply the bisecton method 6 times, which of the following intervals will we end up with?

[2.8125,2.84375]

[2.8125,2.875]

[2.75,2.765625]

[2.6875,2.71875]

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by Brilliant Staff

The equation $\displaystyle 2^x+2^{-x} = 3$ has a root between $1$ and $2.$ If we apply the bisecton method 5 times, which of the following intervals will we end up with?

\left[1.3925, 1.40625\right]

\left[1.4375, 1.5\right]

\left[1.40625, 1.4375\right]

\left[1.375, 1.40625\right]

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by Brilliant Staff