HHG p159 num21


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Without computing the integral, decide if is positive or negative, and explain your decision.

Solution: We know that (in blue) is positive but decreasing. On the other hand, (in green) is positive on , and then repeats itself (but negative) on . So the product (in red) of the two functions will have to be greater (in absolute value) on versus , so the total area will be positive.

hhg_p1589_21

Here is the same graph using desmos.com:

Mathematica source:
f[x_] := Exp[-x];
g[x_] := Sin[x];
h[x_] := f[x]*g[x];
xMin = 0;
xMax = 6.2832;
SetOptions[Plot,
BaseStyle -> {FontFamily -> "Times", FontSize -> 14}, ,
PlotRange -> {{0, xMax}, {-1, 2}}];
p1 = Plot[f[x], {x, 0, xMax}, LabelStyle -> Directive[16],
PlotStyle -> {Blue, Thickness[0.005]}, PlotRange -> {-1.1, 1.1}];
p2 = Plot[g[x], {x, 0, xMax}, LabelStyle -> Directive[16],
PlotStyle -> {Green, Thickness[0.005]}];
p3 = Plot[h[x], {x, 0, xMax}, LabelStyle -> Directive[16],
PlotStyle -> {Red, Thickness[0.005]}];
Show[ p1, p2, p3]

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