Numerical Solution of Equations

By sketching a suitable pair of graphs, show that the equation

has only one root in the interval $0 < x < \pi$.

Show by calculation that this root lies between 1 and 1.5.

Use the iterative formula $x_{n+1} = 2 - 2\sin\frac{1}{2}x_n$ with an initial value of 1.03 to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

By sketching a suitable pair of graphs, show that the equation

has only one root in the interval $0 < x < \frac{1}{2}\pi$.

Show by calculation that this root lies between 0.9 and 1.

Show that if a sequence of values given by the iterative formula

converges, then it converges to the root of the equation in part (a).

Use the iterative formula given in part (c) to calculate $x$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

By sketching a suitable pair of graphs, show that the equation $\cot 2x = 2\sin 2x - 1$ has exactly one root in the interval $0 < x < \frac{1}{2}\pi$.

Show by calculation that the root is in the interval $0.4 < x < 0.6$.

Use the iterative formula $x_{n+1} = \frac{1}{2} \tan^{-1} \left( \frac{1}{2\sin 2x_n - 1} \right)$ to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

Show by calculation that this root lies in the interval $0.7 < x < 0.8$.

Show that if a sequence of values in the interval $0.7 < x < 0.8$ given by the iterative formula

converges then it converges to the root of the equation in part (a).

Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

By sketching a suitable pair of graphs, show that the equation $2 + e^{-0.2x} = \ln(1 + x)$ has only one root.

Show by calculation that this root lies between 7 and 9.

Use the iterative formula

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[$\exp(x)$ is an alternative notation for $e^x$.]

Show that if a sequence of values given by the iterative formula

converges, then it converges to a root of the equation $f(x) = 0$.

The equation has a root close to 1.2.

Use the iterative formula from part (a) and an initial value of 1.2 to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

Show by calculation that $\alpha$ lies between 0.5 and 1.

Show that, if a sequence of positive values given by the iterative formula

converges, then it converges to $\alpha$.

Use this iterative formula to calculate $\alpha$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

By sketching a suitable pair of graphs, show that the equation

has only one root.

Show by calculation that this root lies between 1 and 2.

Show that, if a sequence of values given by the iterative formula

converges, then it converges to the root of the equation in (a).

Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

By sketching a suitable pair of graphs, show that the equation $4 - x^2 = \sec \frac{1}{2}x$ has exactly one root in the interval $0 \le x < \pi$.

Verify by calculation that this root lies between 1 and 2.

Use the iterative formula $x_{n+1} = \sqrt{4 - \sec \frac{1}{2}x_n}$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

By sketching a suitable pair of graphs, show that the equation $\cot \frac{1}{2}x = 1 + e^{-x}$ has exactly one root in the interval $0 < x \le \pi$.

Verify by calculation that this root lies between 1 and 1.5.

Use the iterative formula $x_{n+1} = 2\tan^{-1}\left(\frac{1}{1 + e^{-x_n}}\right)$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.