FPM Graph and Estimate (Chapter 4)

$\let\frac=\dfrac$

1. (2012/jan/paper01/q6) For $x$ radians, $$ y=3 \cos \frac{x}{2} $$

(a) Complete the table, giving the three missing values correct to 2 decimal places. ( 2 marks)
\begin{array}{|l|c|c|c|c|c|c|c|c|} \hline x & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 \\ \hline y & 3 & & 2.63 & 2.20 & & 0.95 & 0.21 & \\ \hline \end{array} On the axes opposite,

(b) draw the graph of $y=3 \cos \frac{x}{2}$ for $0 \leqslant x \leqslant 3.5$ ( 2 marks)

(c) Using your graph, find an estimate, to 1 decimal place, for the root of the equation $$ 2 x=1+2 \cos \frac{x}{2} $$ ( 4 marks)

2. (2013/june/paper02/q7)

(a) Complete the table of values for $y=5 \log _{10}(x+2)-x$, giving your answers to 2 decimal places. \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline y & 1 & 1.51 & 1.39 & & & & -0.77 \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite, draw the graph of $y=5 \log _{10}(x+2)-x$ for $-1 \leqslant x \leqslant 5$ ( 2 marks)

(c) Use your graph to obtain an estimate, to 1 decimal place, of the root of the equation $10 \log _{10}(x+2)-2 x=1 \frac{1}{2}$ in the interval $-1 \leqslant x \leqslant 5$ ( 2 marks)

(d) By drawing an appropriate straight line on your graph, obtain an estimate, to 1 decimal place, of the root of the equation $x=10^{\frac{1}{2} x}-2$ in the interval $-1 \leqslant x \leqslant 5$ ( 4 marks)

3. (2014/jan/paper01/q7)

(a) Complete the table of values for $y=2 x-4+\frac{5}{x^2}$, giving your answers to 2 decimal places where appropriate. \begin{array}{|l|l|l|l|l|l|l|l|l|} \hline x & 0.8 & 1 & 1.5 & 1.7 & 2 & 2.5 & 3 & 4 \\ \hline y & 5.41 & & 1.22 & & & 1.8 & & 4.31 \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite, draw the graph of $y=2 x-4+\frac{5}{x^2}$ for $0.8 \leqslant x \leqslant 4$ ( 2 marks)

(c) Use your graph to obtain estimates, to 1 decimal place, of the roots of the equation $2 x+\frac{5}{x^2}=6$ in the interval $0.8 \leqslant x \leqslant 4$ ( 2 marks)

(d) By drawing a straight line on your graph obtain an estimate, to 1 decimal place, of the root of the equation $4 x+\frac{5}{x^2}=12$ in the interval $0.8 \leqslant x \leqslant 4$ ( 4 marks)

4. (2015/jan/paper02/q5)
The grid opposite shows the graph of $y=3 x \sin x$ for $-1 \leqslant x \leqslant 3$, where $x$ is measured in radians.

(a) Use the graph to estimate, to 1 decimal place, the roots of the equation $$ x \sin x=1 $$ in the interval $-1 \leqslant x \leqslant 3$ ( 3 marks)

(b) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, of the roots of the equation $2 x \sin x-x=1$ in the interval $-1 \leqslant x \leqslant 3$ ( 5 marks)

5. (2015/june/paper02/q2)

(a) Complete the table of values for $y=x+\frac{6}{x^2}$ Give your answers to 2 decimal places where necessary. \begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline x & 1.0 & 1.25 & 1.5 & 1.75 & 2.0 & 2.25 & 2.5 & 2.75 & 3.0 \\ \hline y & & & 4.17 & 3.71 & & 3.44 & & 3.54 & 3.67 \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite, draw the graph of $y=x+\frac{6}{x^2}$ for $1 \leqslant x \leqslant 3$ ( 2 marks)

(c) By drawing a suitable straight line on the grid, obtain estimates, to 1 decimal place, for the solutions of the equation $x^3-3 x^2+3=0$ in the interval $1 \leqslant x \leqslant 3$ ( 4 marks)

6. (2016/june/paper01/q7)

(a) Complete the table of values for $y=2^x-4$, giving your answers to 2 decimal places. \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 2.75 & 3 \\ \hline y & -3 & & -2 & & 0 & & 2.73 & 4 \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite, draw the graph of $y=2^x-4$ for $0 \leqslant x \leqslant 3$ ( 2 marks)

(c) Use your graph to obtain an estimate, to one decimal place, of the value of $\log _2 7$ Show clearly how you used the graph. ( 3 marks)

(d) By drawing a straight line on your graph, obtain an estimate to one decimal place of the root of the equation $2^x+3 x=7$ in the interval $0 \leqslant x \leqslant 3$ ( 4 marks)

7. (2017/jan/paper01/q7)

(a) Complete the table of values for $y=\ln (5 x+1)+2$ giving your answers to 2 decimal places. \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline y & 2 & & 4.40 & 4.77 & 5.04 & & 5.43 & \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite draw the graph of $y=\ln (5 x+1)+2$ for $0 \leqslant x \leqslant 7$ ( 2 marks)

(c) By drawing an appropriate straight line on the grid, obtain an estimate, to 1 decimal place, of the positive root of the equation $\ln (5 x+1)-x=0$ in the interval $0 \leqslant x \leqslant 7$ ( 3 marks)

(d) By drawing an appropriate straight line on the grid, obtain an estimate, to 1 decimal place, of the root of the equation $\mathrm{e}^{(3 x-1)}=5 x+1$ in the interval $0 \leqslant x \leqslant 7$ ( 4 marks)

8. (2018/jan/paper01/q5)

(a) Complete the table of values for $y=\frac{x^3+2}{x+1}$ giving your answers to 2 decimal places where appropriate. \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 0.5 & 1 & 1.5 & 2 & 3 & 4 \\ \hline y & & 1.42 & & 2.15 & & 7.25 & \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite draw the graph of $y=\frac{x^3+2}{x+1}$ for $0 \leqslant x \leqslant 4$ ( 2 marks)

(c) By drawing a suitable straight line on your graph obtain an estimate, to 1 decimal place, of the root of the equation $x^3+x^2-3 x-2=0$ in the interval $0 \leqslant x \leqslant 4$ ( 5 marks)

9. (2018/june/paper01/q4)
Figure 2 shows the graph of $y=x-\frac{1}{2 x^2}$ for $0.4 \leqslant x \leqslant 5$ drawn on a grid.

(a) (i) Express $x-\frac{1}{2 x^2}$ as a single fraction. (ii) Hence use the graph to obtain, to one significant figure, an estimate for the value of $\sqrt[3]{0.5}$ ( 3 marks)

(b) By drawing a suitable straight line on the grid, find an estimate to 2 significant figures, for the root of the equation $$ 4-2 x+\frac{1}{2 x^2}=0 $$ in the interval $0.4 \leqslant x \leqslant 5$ ( 3 marks)

10. (2019/juneR/paper01/q8)

(a) Complete the table of values for $y=2+\ln (2 x+1)$ giving your answers to 2 decimal places. ( 2 marks)
\begin{array}{|c|c|c|c|c|c|c|c|} \hline x & 0 & 0.25 & 0.5 & 1 & 1.5 & 2 & 3 \\ \hline y & 2 & & & 3.10 & 3.39 & 3.61 & \\ \hline \end{array}

(b) On the grid opposite, draw the graph of $y=2+\ln (2 x+1)$ for $0 \leqslant x \leqslant 3$ ( 2 marks)

(c) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation $\ln (2 x+1)=3 x-4$ in the interval $0 \leqslant x \leqslant 3$ ( 3 marks)

(d) By drawing an appropriate straight line on the grid, obtain an estimate, to one decimal place, of the root of the equation $e^{(6-x)}-(2 x+1)^2=0$ in the interval $0 \leqslant x \leqslant 3$ ( 4 marks)

11. (2016/jan/paper02/q11)

(a) Complete the table of values for $y=\mathrm{e}^{(x-1)}+2$ Give your answers to 2 decimal places where appropriate. \begin{array}{|c|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \mathrm{f}(x) & 2.05 & & & & 4.72 & 9.39 \\ \hline \end{array} ( 2 marks)

(b) On the grid opposite, draw the graph of $y=\mathrm{e}^{(x-1)}+2$ for $-2 \leqslant x \leqslant 3$ ( 2 marks)

(c) Use your graph to obtain an estimate, to 1 decimal place, of the root of the equation $4=\mathrm{e}^{(x-1)}$ in the interval $-2 \leqslant x \leqslant 3$ ( 2 marks)

(d) By drawing a straight line on the grid, obtain an estimate, to I decimal place, of the root of the equation $\ln (4 x-4)=x-1$ in the interval $-2 \leqslant x \leqslant 3$ ( 5 marks)

 



Answer

1. (a) $\quad 2.91,1.62,-0.53$ (b) Graph (c) $x=1.3$

2. (a) $\quad 1.01,0.49,-0.11$ (b) Graph (c) $x=2.5$ (d) $x=0.9$

3. (a) $\quad 3,1.13,1.25,2.56$ (b) Graph(c) $x=1.2,2.6$ (d) $x=2.8$

4. (a) $x=1.1,2.8$ (b) Graph, $x=-0.5,1.2,2.3$

5. (a) $7,5.09,3 \cdot 5,3.46$ (b) Graph (c) $y=6-x, x=1.3,2.5$

6. (a) $-2.59,-1.17,1.66$ (b) Graph (c) $y=3,2 \cdot 8$ (d) $y=3-3 x, x=1.4$

7. (a) $3.79,5.26, 5.58$ (b) Figure (c) $y=x+2$ $x=2.6$ (d) $y=3x+1,x=0.9$

8. (a) $2,1.5,3.33,13.2$ (b) Graph (c) $y=-x+4,x=1.6$

9. (a)(i) $\frac{2x^3-1}{2x^2}$ (ii) $x\approx 0.8$ (b) $y=4-x$, $x=2.1$

10. (a) $2.41,2.69,3.95$ (b) Graph (c) Graph $y=3x-2,x=1.8$ (d) Graph $y=5-\frac x2,x=2.4$

11. (a) $2 \cdot 14,2 \cdot 37,3$ (b) Graph(c) $y=6, \quad x=2.4$ (d) $y=4 x-2, x=1.3$


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