Visualizing the Product of Two Functions

\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \newcommand{\tr}{{\rm tr\,}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \renewcommand{\AA}{\vf A} \newcommand{\BB}{\vf B} \newcommand{\CC}{\vf C} \newcommand{\EE}{\vf E} \newcommand{\FF}{\vf F} \newcommand{\GG}{\vf G} \newcommand{\HH}{\vf H} \newcommand{\II}{\vf I} \newcommand{\JJ}{\vf J} \newcommand{\KK}{\vf K} \renewcommand{\SS}{\vf S} \renewcommand{\aa}{\VF a} \newcommand{\bb}{\VF b} \newcommand{\ee}{\VF e} \newcommand{\gv}{\VF g} \newcommand{\iv}{\vf\imath} \newcommand{\rr}{\VF r} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF u} \newcommand{\vv}{\VF v} \newcommand{\ww}{\VF w} \newcommand{\grad}{\vf\nabla} \newcommand{\zero}{\vf 0} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\LL}{\mathcal{L}} \newcommand{\DD}[1]{D_{\textrm{$#1$}}} \newcommand{\bra}[1]{\langle#1|} \newcommand{\ket}[1]{|#1/rangle} \newcommand{\braket}[2]{\langle#1|#2\rangle} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\ii}{\ihat} \newcommand{\jj}{\jhat} \newcommand{\kk}{\khat} \newcommand{\dS}{dS} \newcommand{\dA}{dA} \newcommand{\dV}{d\tau} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 4.5 Visualizing the Product of Two Functions

The applet below shows the graph of the product $fg$ (in green) of two functions $f$ (in brown) and $g$ (in blue). ¹ This graph can be constructed by pointwise multiplication: At each point in the domain, read off the values of the two functions from their graphs, multiply these values together, and plot that value.

Figure 4.5.1. The product of two functions.

Activity 4.5.1. Multiplying Functions Together.

Try it for yourself! Choose any two functions and graph them on the same axes. Looking only at these graphs, construct the graph of the product of the two functions. After completing this geometric construction, multiply the two functions together and graph the result. Did you draw the correct graph?

Hint.

You can check your answer by entering the functions into the applet in Figure 4.5.1.

The integral of the product is also shown; see Section 10.3.