Using your Graphic Display Calculator (GDC)

Calculating with numbers

1. Calculate the following calculations with your Graphic Display Calculator (GDC). Write your results as integers or fractions.
   

   (a)   $1+(-2\cdot 3+8)\cdot 5-4$

   (b)   ${12}^3-13\cdot 3^4$

   (c)   $-\frac{2}{3}(5-(-2)(-6))$

   (d)   $\frac{11}{9}+\frac{9}{8}\cdot \frac{4}{3}$

   (e)   ${\left(\frac{5}{2}\right)}^3-3\cdot 5$

   Solutions:    (a)  $7$;     (b)  $675$;     (c)  $\frac{14}{3}$;     (d)  $\frac{49}{18}$;     (e)  $\frac{5}{8}$
2. Calculate the following calculations with your GDC. Write your results as mixed numbers.
   

   (a)   $4\frac{5}{6}+2\frac{2}{3}$

   (b)   $4\cdot \frac{5}{6}+2\cdot \frac{2}{3}$

   (c)   ${\displaystyle \frac{1+\frac{5}{3}}{1-\frac{3}{5}}}$

   (d)   $\sqrt{14\cdot (1\frac{1}{2}\cdot \frac{1}{15}+\frac{1}{25})}$

   Hint:    Maybe you'll have to type  $4\frac{5}{6}$  as  $(4+\frac{5}{6})$.
   Solutions:    (a)  $7\frac{1}{2}$;     (b)  $4\frac{2}{3}$;     (c)  $6\frac{2}{3}$;     (d)  $1\frac{2}{5}$
3. Calculate the following calculations with your GDC. Write your results as decimals, rounding them to three significant figures.
   

   (a)   $\frac{1}{7}\cdot (\frac{5}{3}-\frac{1}{6})$

   (b)   $\frac{5\cdot (-2)+17\cdot 3}{13}$

   (c)   $\frac{1234}{3\cdot 7-6}$

   (d)   ${\left(\frac{3}{13}\right)}^3$

   (e)   $\frac{2000-\frac{8}{7}}{{50}^2}$

   Solutions:    (a)  $0.214$;     (b)  $3.15$;     (c)  $82.3$;     (d)  $0.0123$;     (e)  $0.800$
4. Calculate the following calculations with your GDC. Round your results to three significant figures. Use scientific notation: $a\times {10}^n$.
   

   (a)   $\frac{{4321}^2}{3\cdot 4-7}$

   (b)   ${\left({135}^3\right)}^7$

   (c)   $\frac{2.73\times {10}^{37}}{3.81\times {10}^{13}}$

   (d)   ${(2.713\cdot {10}^5)}^3$

   Solutions:    (a)  $3.73\cdot {10}^6$;     (b)  $5.46\cdot {10}^{44}$;     (c)  $7.17\cdot {10}^{23}$;     (d)  $2.00\cdot {10}^{16}$
5. ?
   ?

   Rounding to a low number of decimals and executing further calculations with approximate values can result in a totally incorrect final result.
   
   To get a correct result you must always store your results in the calculator's memory. In further calculations you must use the stored values.
   In following calculations use your GDC and write down your results rounded to three significant figures.

   (a)   Calculate the number $a=\frac{12+\sqrt{5}}{3+\sqrt{2}}$.

   (b)   After calculating store the obtained value into your calculator's memory.

   (c)   Use the stored value to calculate $m=\frac{40a-130}{31a-100}$.

   Consider other possibilities of calculating $m$. Instead of using the value stored in your GDC's memory, you could use the approximate value for $a$. Do it and explain the obtained results.

   (d)   Use $a\approx 3.23$ and calculate $m$.

   (e)   Use $a\approx 3.225$ and calculate $m$.

   (f)   Use $a\approx 3.22505$ and calculate $m$.

   Solutions:    (a)  $a\approx 3.23$;     (c)  $m\approx 42.7$;     (d)  $m\approx -6.15$ (!);     (e)  $m\approx 40$;     (f)  $m\approx 42.6$
   If you want to get $m$ correct to three significant figures you must use much more figures in $a$. The best variant is using the stored value.
6. Calculate numbers $x,y$ and $z$. Store the values into your calculator's memory and use them in later calculations. Write the final results rounded to three significant figures.
   

   (a)   $x=\frac{14}{14\sqrt{7}-37}$

   (b)   $y=\frac{x-123}{x+321}$

   (c)   $z={\left(\frac{1-y}{x+y+3}\right)}^2$

   Solutions:    (a)  $x\approx 346$;     (b)  $y\approx 0.334$;     (c)  $z\approx 3.65\cdot {10}^{-6}$
7. Given the numbers $a=0.0022349$,  $b=\sqrt{\frac{12}{3456789}}$ and $c=1.5867\cdot {10}^7$ calculate the following calculations using your GDC and write the results rounded to three significant figures.
   

   (a)   $abc$

   (b)   $ab{c}^{-1}$

   (c)   $(abc{)}^{-1}$

   (d)   $(a-b)c$

   Hint:    First store the given numbers into your calculator's memory.
   Solutions:    (a)  $abc\approx 66.1$;     (b)  $ab{c}^{-1}\approx 2.62\cdot {10}^{-13}$;     (c)  $(abc{)}^{-1}\approx 0.0151$ or $1.51\cdot {10}^{-2}$;     (d)  $(a-b)c\approx 5.90\cdot {10}^3$

Drawing graphs

8. Function $f$ has the equation $f(x)=\frac{1}{2}x-3$.
   

   (a)   Draw the graph of $f$.

   (b)   Determine the axis intercepts.

   Solutions:    (b)  $A(6,0),B(0,-3)$
9. Function $f$ has the equation $f(x)=\frac{3}{2}x+4$.
   

   (a)   Draw the graph of $f$.

   (b)   Determine the axis intercepts.

   (c)   Calculate the area of the triangle formed by this function and the coordinate axes.

   Solutions:    (b)  $P_1(-2.67,0),P_2(0,4)$;     (c)  $A\approx 5.33$
10. Consider the following two straight lines:  $y=3x-2,y=\frac{1}{2}x+3$
    

    (a)   Find the intersection point by solving the appropriate equation.

    (b)   Find the intersection point using the Analyze Graph function on your GDC.

    Solution:    $P(2,4)$
11. Straight lines ${\ell }_1$ and ${\ell }_2$ have the equations    ${\ell }_1:y=-2x+6$    and    ${\ell }_2:y=\frac{1}{2}x-1$.
    

    (a)   Find the point of intersection $P$.

    (b)   Find the area of the triangle formed by the lines ${\ell }_1,{\ell }_2$ and $y$-axis.

    Solutions:    (a)  $P(\frac{14}{5},\frac{2}{5})$;     (b)  $A=\frac{49}{5}=9.8$
12. Solve the following systems of simultaneous linear equations.
    First draw both graphs and then use the Analyze Graph function to find the intersection point.
    

    (a)   $y=2x+3,y=6-x$

    (b)   $y=\frac{3}{2}x+1,y=3x-4$

    (c)   $2x+y=7,x-4y+8=0$

    Solutions:    (a)  $x=1,y=5$;     (b)  $x\approx 3.33,y=6$;     (c)  $x\approx 2.22,y\approx 2.56$

Solving equations

13. Solve the following linear equations.
    First draw graphs for the left-hand side and for the right-hand side. Then use the Analyze Graph function to find the intersection point.
    

    (a)   $3x-1=x+13$

    (b)   $5x-3=6-2x$

    (c)   $x-\frac{x+4}{3}=\frac{5x+1}{6}-1$

    Solutions:    (a)  $x=7$;     (b)  $x\approx 1.29$;     (c)  $x=-3$
14. Solve the following non-linear equations.
    First draw graphs for the left-hand side and for the right-hand side. Then use the Analyze Graph function to find the intersection point.
    

    (a)   $x^3=3-2x$

    (b)   $x-1=\sqrt{x+4}$

    (c)   $\frac{1}{x}=\frac{x-1}{2}$

    Solutions:    (a)  $x=1$;     (b)  $x\approx 3.79$;     (c)  two solutions: $x_1=-1,x_2=2$
15. Solve the following systems of simultaneous linear equations.
    You can solve them using the Analyze Graph function. You can also try the LinSolve function (it doesn't require graphs).
    

    (a)   $x+2y=10,3x+5y=27$

    (b)   $y=2x+7,4x+2y=6$

    (c)   $7x-y=5,5x+3y=-2$

    (d)   $\frac{2}{3}x+\frac{4}{3}y=\frac{2}{3},\frac{1}{2}x+\frac{3}{2}y=-1$

    Solutions:    (a)  $x=4,y=3$;     (b)  $x=-1,y=5$;     (c)  $x=\frac{1}{2},y=-\frac{3}{2}$;     (d)  $x=7,y=-3$

Modelling

16. In the USA the temperature is measured in degrees Farenheit (°F). The rest of the world uses degrees Celsius (°C), called also centigrades. The freezing point of water is at 32 °F and the boiling point is at 212 °F.
    

    (a)   Write the linear function $y=mx+c$ which converts the given temperature $x$ °C  to  $y$ °F.

    (b)   Draw the graph of this function. Use appropriate units.

    (c)   Determine which temperature (in °C) corresponds to 0 °F.

    (d)   Which temperature corresponds to the same number of degrees in both scales?

    Solutions:    (a)  $y=1.8x+32$;     (c)  $x\approx -17.8$ °C;     (d)  −40 °C = −40 °F
17. Mary's Grocery Shop sells potatoes at two different prices:
    – if you buy less than 10 kg, the price is 1.20 € per kg,
    – if you buy 10 kg or more, the price is 0.90 € per kg.
    

    (a)   Draw the graph of the function which represent the cost of $x$ kilograms of potatoes.

    (b)   Explain why buying 9 kg doesn't make sense.

    (c)   Find all other quantities $x$, you'd rather not buy.

    Solutions:    (a)  Use appropriate units!     (b)  It costs more than 10 kg.     (c)  $7.5<x<10$
18. You would like to have a cappuccino with some friends. You can chose between two possibilities:
    – at Alpha Caffe one cappuccino costs 2 €,
    – at Beta Caffe one cappuccino costs 1.50 €, but you must pay 3 € extra for the table reservation.
    

    (a)   Draw the graph of the function $f(x)$ which represents the cost of $x$ cappuccinos at Alpha Caffe.

    (b)   Draw the graph of the function $g(x)$ which represents the cost of $x$ cappuccinos at Beta Caffe.

    (c)   Find the intersection point and explain its meaning. Which caffe would you chose if you want to pay less?

    Solutions:    (a,b)  Use appropriate units!     (c)  $P(6,12)$

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