(a) \(1+(-2\cdot 3+8)\cdot 5-4\)
(b) \(12^3-13\cdot3^4\)
(c) \(-\frac{\textstyle 2}{\textstyle 3}(5-(-2)(-6))\)
(d) \(\frac{\textstyle 11}{\textstyle 9}+\frac{\textstyle 9}{\textstyle 8}\cdot\frac{\textstyle 4}{\textstyle 3}\)
(e) \(\left(\frac{\textstyle 5}{\textstyle 2}\right)^3-3\cdot5\)
Solutions: (a) \(7\); (b) \(675\); (c) \(\frac{14}{3}\); (d) \(\frac{49}{18}\); (e) \(\frac{5}{8}\)(a) \(4\frac{\textstyle 5}{\textstyle 6}+2\frac{\textstyle 2}{\textstyle 3}\)
(b) \(4\cdot\frac{\textstyle 5}{\textstyle 6}+2\cdot\frac{\textstyle 2}{\textstyle 3}\)
(c) \({\displaystyle\frac{1+\frac{5}{3}}{1-\frac{3}{5}}}\)
(d) \(\sqrt{14\cdot\left(1\frac{\textstyle 1}{\textstyle 2}\cdot\frac{\textstyle 1}{\textstyle 15} +\frac{\textstyle 1}{\textstyle 25}\right)}\)
Hint: Maybe you'll have to type \(4\frac{5}{6}\) as \((4+\frac{5}{6})\).(a) \(\frac{\textstyle 1}{\textstyle 7}\cdot\left(\frac{\textstyle 5}{\textstyle 3}-\frac{\textstyle 1}{\textstyle 6}\right)\)
(b) \(\frac{\textstyle 5\cdot(-2)+17\cdot3}{\textstyle 13}\)
(c) \(\frac{\textstyle 1234}{\textstyle 3\cdot7-6}\)
(d) \(\left(\frac{\textstyle 3}{\textstyle 13}\right)^3\)
(e) \(\frac{\textstyle 2000-\frac{8}{7}}{\textstyle 50^2}\)
Solutions: (a) \(0.214\); (b) \(3.15\); (c) \(82.3\); (d) \(0.0123\); (e) \(0.800\)(a) \(\frac{\textstyle 4321^2}{\textstyle 3\cdot4-7}\)
(b) \(\left(135^3\right)^7\)
(c) \(\frac{\textstyle 2.73\times10^{37}}{\textstyle 3.81\times10^{13}}\)
(d) \(\left(2.713\cdot10^5\right)^3\)
Solutions: (a) \(3.73\cdot10^6\); (b) \(5.46\cdot 10^{44}\); (c) \(7.17\cdot10^{23}\); (d) \(2.00\cdot10^{16}\)(a) Calculate the number \(a=\frac{\textstyle 12+\sqrt{5}}{\textstyle 3+\sqrt{2}}\).
(b) After calculating store the obtained value into your calculator's memory.
(c) Use the stored value to calculate \(m=\frac{\textstyle 40a-130}{\textstyle 31a-100}\).
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 \(a\approx3.23\) and calculate \(m\).
(e) Use \(a\approx3.225\) and calculate \(m\).
(f) Use \(a\approx3.22505\) and calculate \(m\).
Solutions: (a) \(a\approx3.23\); (c) \(m\approx42.7\); (d) \(m\approx-6.15\) (!); (e) \(m\approx40\); (f) \(m\approx42.6\)(a) \(x=\frac{\textstyle 14}{\textstyle 14\sqrt{7}-37}\)
(b) \(y=\frac{\textstyle x-123}{\textstyle x+321}\)
(c) \(z=\left(\frac{\textstyle 1-y}{\textstyle x+y+3}\right)^2\)
Solutions: (a) \(x\approx346\); (b) \(y\approx0.334\); (c) \(z\approx3.65\cdot10^{-6}\)(a) \(abc\)
(b) \(abc^{-1}\)
(c) \((abc)^{-1}\)
(d) \((a-b)c\)
Hint: First store the given numbers into your calculator's memory.(a) Draw the graph of \(f\).
(b) Determine the axis intercepts.
Solutions: (b) \(A(6,0),~ B(0,-3)\)(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) \(A\approx5.33\)(a) Find the intersection point by solving the appropriate equation.
(b) Find the intersection point using the Analyze Graph function on your GDC.
Solution: \(P(2,4)\)(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) \(A=\frac{49}{5}=9.8\)(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) \(x\approx3.33,~ y=6\); (c) \(x\approx2.22,~ y\approx2.56\)(a) \(3x-1=x+13\)
(b) \(5x-3=6-2x\)
(c) \(x-\frac{\textstyle x+4}{\textstyle 3}=\frac{\textstyle 5x+1}{\textstyle 6}-1\)
Solutions: (a) \(x=7\); (b) \(x\approx1.29\); (c) \(x=-3\)(a) \(x^3=3-2x\)
(b) \(x-1=\sqrt{x+4}\)
(c) \(\frac{\textstyle 1}{\textstyle x}=\frac{\textstyle x-1}{\textstyle 2}\)
Solutions: (a) \(x=1\); (b) \(x\approx3.79\); (c) two solutions: \(x_1=-1,~ x_2=2\)(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) \(x=7,~ y=-3\)(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(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) \(7.5\lt x\lt 10\)(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) \(P(6,12)\)