Free Printable Chapter 3 Numerical Methods I 3.1-3.3

1.

By using graphical method, there is a real root between [a,b] for $\ln\left(x-2\right)+x^2-4=0$ Hence, state the values of a and b.

a)

$a=1,\ b=4$

b)

$a=2,\ b=3$

c)

$a=0,\ b=2$

d)

$a=0,\ b=4$

2.

Is there a real root for $x^2-\sqrt[]{x}-36=0$ between [6, 6.5]?

a)

Yes

b)

No

3.

Show that there is a real root for $x^3+1=x^2-9$ by sketching graphs.

State its approximate value.

a)

$x_0=1$

b)

$x_0=-4$

c)

$x_0=-2$

d)

$x_0=\frac{1}{2}$

4.

The equation $e^x=4\sin x$ has a root between x=1 and x=2. Hence, by using Newton-Raphson method, find the real root correct to 3 s.f.

a)

$2.05$

b)

$1.78$

c)

$2.61$

d)

$1.37$

5.

By taking 0.2 as the first approximation, evaluate the real root of the equation $x^2-\frac{1}{x}+4=0$ correct to 3 s.f.

a)

0.246

b)

0.635

c)

0.153

d)

0.724

6.

Show that the equation $2x^3+x^2=33$ has a root in the interval $2<x<2.5$ .

Find this root correct to 3 s.f.

a)

2.17

b)

2.45

c)

2.39

d)

2.28

7.

By taking $x=2$ as the first approximation, calculate using Newton-Raphson method, the third approximation to $7^{\frac{1}{3}}$ (3 s.f.)

a)

1.91

b)

1.92

c)

1.89

d)

1.90

8.

Estimate $\int_0^8\sqrt[]{1+x^2}dx$ by using trapezoidal rule with 5 ordinates correct to 3 d.p. What is this value?

a)

33.946; approximated value

b)

33.946; absolute value

c)

32.246; approximated value

d)

32.246; absolute value

9.

Evaluate $\int_0^1\ \left(6xe^{3x^2+1}\right)dx$ correct to 4 d.p.

(Note: We'll be using this value later in Q11)

a)

52.4290

b)

51.8799

c)

51.8273

d)

52.0164

10.

Estimate $\int_0^1\ \left(6xe^{3x^2+1}\right)dx$ by using trapezoidal rule with 5 subintervals correct to 4 d.p.

(Note: We'll be using this value later in Q11)

a)

59.1659

b)

57.7352

c)

58.0016

d)

56.7529

11.

By using the answers found from Q9 and Q10, compute the error.

a)

7.2861

b)

5.9253

c)

7.9014

d)

6.3881

12.

Use the trapezoidal rule to estimate $\int_0^2g\left(x\right)dx$ from the data given:

a)

13.9962

b)

16.9473

c)

17.4530

d)

15.8836

Chapter 3 Numerical Methods I 3.1-3.3

Total questions: 12

Worksheet time: 2hrs 0mins

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