1. Belinda loves chocolate and always thinks that more is better than less. Belinda thinks that a few piano lessons would be worse than none at all, but if she had enough piano lessons to get good at playing the
piano, she would prefer more lessons to less. Draw a graph with piano lessons on the horizontal axis and
chocolate on the vertical axis. On your graph sketch two indifference curves for Belinda that would be
consistent with this story. Label the better of the two indifference curves AA and the worse one BB.

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

Efficient homemakers ltd. makes canvas wallets and leather wallets as part of a money-making project. for the canvas wallets, they need two yards of canvas and two yards of leather. for the leather wallets, they need four yards of leather and three yards of canvas. their production unit has purchased 44 yards of leather and 40 yards of canvas. let x be the number of leather wallets and y be the number of canvas wallets. draw the graph showing the feasible region to represent the number of the leather and canvas wallets that can be produced.

If jklm is a trapezoid, which statements must be true?