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O gráfico da função \(f\), definida por \(f(x)=\sin {x}\), e dois pontos do gráfico de \(f\), \(P\) e \(Q\), são dados na
figura abaixo.
![[Picture]](/rendered/rulesOfDifferentiation/exercises/derivativeIntersectionOfTangentLines-bf3e34e9763346afd4670a490e062452.svg)
-
Determine uma equação da reta tangente à curva \(y=f(x)\)
no ponto \(P\).
Primeiro, as coordenadas do ponto \(P\) são \[ P=(0,\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}}). \] Para encontrar uma equação
dessa reta tangente, precisamos determinar a inclinação, \(m\). \[ m=f'(\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}})=\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}}, \] já que \[ f'(x)=\class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}}. \]
Portanto, a equação da reta tangente à curva \(y=f(x)\) no ponto \(P\) é dada por \[ y=\class{xm-answer}{\cssId{xm-answer-5}{\phantom{\rule{7em}{1.5em}}}}x. \]
-
Determine uma equação da reta tangente à curva \(y=f(x)\)
no ponto \(Q\).
Primeiro, as coordenadas do ponto \(Q\) são \[ Q=(\pi ,\class{xm-answer}{\cssId{xm-answer-6}{\phantom{\rule{7em}{1.5em}}}}). \]
Para encontrar uma equação dessa reta tangente, precisamos determinar
a inclinação, \(m\). \[ m=f'(\class{xm-answer}{\cssId{xm-answer-7}{\phantom{\rule{7em}{1.5em}}}})=\class{xm-answer}{\cssId{xm-answer-8}{\phantom{\rule{7em}{1.5em}}}}. \] Portanto, a equação da reta tangente à curva \(y=f(x)\) no ponto \(Q\)
é dada por \[ y=\class{xm-answer}{\cssId{xm-answer-9}{\phantom{\rule{7em}{1.5em}}}}(x-\pi ). \]
-
As duas retas tangentes, definidas no item (a) e no item (b), são mostradas
na figura abaixo.
Determine o ponto de interseção dessas duas retas ou mostre que tal ponto
NÃO existe.
Para encontrar o ponto de interseção dessas duas retas tangentes, temos que
resolver o seguinte sistema de equações: \[ y=\class{xm-answer}{\cssId{xm-answer-10}{\phantom{\rule{7em}{1.5em}}}}x \] \[ y=\class{xm-answer}{\cssId{xm-answer-11}{\phantom{\rule{7em}{1.5em}}}}(x-\pi ). \] A solução desse sistema de equações
e, portanto, o ponto de interseção das duas retas tangentes é o ponto
\[ (\class{xm-answer}{\cssId{xm-answer-12}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-13}{\phantom{\rule{7em}{1.5em}}}}). \]
![[Picture]](/rendered/rulesOfDifferentiation/exercises/derivativeIntersectionOfTangentLines-ea50fa2dec5c8d5a6d9ebca6f8f00927.svg)