\(
\newenvironment {prompt}{}{}
\newcommand {\accordion }[0]{}
\newcommand {\ungraded }[0]{}
\newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex}
\newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty}
\newcommand {\todo }[0]{}
\newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint }
\newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}}
\newcommand {\tkzTabSlope }[1]{\foreach \x /\y /\z in {#1}{\node [left = 3pt] at (Z\x 1) {\scriptsize $\y $}; \node [right = 3pt] at (Z\x 1) {\scriptsize $\z $}; }}
\newcommand {\RR }[0]{\mathbb R}
\newcommand {\R }[0]{\mathbb R}
\newcommand {\N }[0]{\mathbb N}
\newcommand {\Z }[0]{\mathbb Z}
\newcommand {\sagemath }[0]{\textsf {SageMath}}
\newcommand {\d }[0]{\,d}
\newcommand {\l }[0]{\ell }
\newcommand {\ddx }[0]{\frac {d}{\d x}}
\newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}}
\newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}}
\newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}}
\newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}}
\newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty - \infty }}}
\newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}}
\newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}}
\newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}}
\newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}}
\newcommand {\dfn }[0]{\textbf }
\newcommand {\unit }[0]{\mathop {}\!\mathrm }
\newcommand {\eval }[1]{\bigg [ #1 \bigg ]}
\newcommand {\seq }[1]{\left ( #1 \right )}
\newcommand {\epsilon }[0]{\varepsilon }
\newcommand {\phi }[0]{\varphi }
\newcommand {\iff }[0]{\Leftrightarrow }
\DeclareMathOperator {\arccot }{arccot}
\DeclareMathOperator {\arcsec }{arcsec}
\DeclareMathOperator {\arccsc }{arccsc}
\DeclareMathOperator {\si }{Si}
\DeclareMathOperator {\scal }{scal}
\DeclareMathOperator {\sign }{sign}
\newcommand {\arrowvec }[1]{{\overset {\rightharpoonup }{#1}}}
\newcommand {\vec }[1]{{\overset {\boldsymbol {\rightharpoonup }}{\mathbf {#1}}}\hspace {0in}}
\newcommand {\point }[1]{\left (#1\right )}
\newcommand {\pt }[1]{\mathbf {#1}}
\newcommand {\Lim }[2]{\lim _{\point {#1} \to \point {#2}}}
\DeclareMathOperator {\proj }{\mathbf {proj}}
\newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}}
\newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}}
\newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}}
\newcommand {\vecl }[0]{\vec {\boldsymbol {\l }}}
\newcommand {\uvec }[1]{\mathbf {\hat {#1}}}
\newcommand {\utan }[0]{\mathbf {\hat {t}}}
\newcommand {\unormal }[0]{\mathbf {\hat {n}}}
\newcommand {\ubinormal }[0]{\mathbf {\hat {b}}}
\newcommand {\dotp }[0]{\bullet }
\newcommand {\cross }[0]{\boldsymbol \times }
\newcommand {\grad }[0]{\boldsymbol \nabla }
\newcommand {\divergence }[0]{\grad \dotp }
\newcommand {\curl }[0]{\grad \cross }
\newcommand {\lto }[0]{\mathop {\longrightarrow \,}\limits }
\newcommand {\bar }[0]{\overline }
\newcommand {\surfaceColor }[0]{violet}
\newcommand {\surfaceColorTwo }[0]{redyellow}
\newcommand {\sliceColor }[0]{greenyellow}
\newcommand {\vector }[1]{\left \langle #1\right \rangle }
\newcommand {\sectionOutcomes }[0]{}
\newcommand {\VerbMathToks }[2]{\alteqtoks {\begin {#2} #1 \end {#2}}}
\newcommand {\DeclareMicrotypeVariants }[1]{}
\newcommand {\DeclareMicrotypeAlias }[2]{}
\newcommand {\LoadMicrotypeFile }[1]{}
\newcommand {\DeclareMicrotypeFilePrefix }[1]{}
\newcommand {\DeclareMicrotypeBabelHook }[2]{}
\newcommand {\microtypesetup }[1]{}
\newcommand {\microtypecontext }[1]{}
\newcommand {\textmicrotypecontext }[2]{#2}
\newcommand {\leftprotrusion }[1]{#1}
\newcommand {\rightprotrusion }[1]{#1}
\newcommand {\noprotrusionifhmode }[0]{}
\newcommand {\lsstyle }[0]{}
\newcommand {\lslig }[1]{#1}
\newcommand {\maketitle }[0]{}
\newcommand {\problem }[0]{ }
\newcommand {\exercise }[0]{ }
\newcommand {\question }[0]{ }
\newcommand {\exploration }[0]{ }
\newcommand {\dialogue }[0]{\begin {description}}
\newcommand {\GetTranslation }[1]{#1}
\)
Seja \(s(t) = t^3 - 2t^2\). O gráfico de \(s\) é dado abaixo.
Suponha que \(A\) e \(B\) sejam dois pontos no gráfico de \(s\). Preencha as lacunas.
\begin{align*} A &= \left (2,\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}}\right )\\ B &= \left (1,\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}}\right ) \end{align*}
Marcamos \(A\) e \(B\) no gráfico de \(s\) abaixo.
A inclinação da reta secante entre \(A\) e \(B\) é \(\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}}\).
Desenhamos a reta secante entre \(A\) e \(B\) e também a reta tangente a \(s\) em \(A\).
Com base no desenho da reta tangente, um bom número inteiro para estimar a
inclinação da reta tangente é \(\class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}}\)
Suponha que a função \(s(t) = t^3 - 2t^2\) represente a posição, no instante \(t\), de um objeto que se move
ao longo de uma reta, em que \(s\) é medida em pés e \(t\) é medido em segundos. A
velocidade média do objeto no intervalo de tempo \(\left [1,2\right ]\) é \(\class{xm-answer}{\cssId{xm-answer-5}{\phantom{\rule{7em}{1.5em}}}}\)\(ft/s\).
Complete a tabela a seguir com as velocidades médias apropriadas (em \(ft/s\)),
arredondadas para duas casas decimais.
\[ \begin{array}{|c|c|c|c|c|c|} \hline \text {Time interval} & \left [1,2\right ] & \left [1.5,2\right ] & \left [1.9,2\right ] & \left [2,2.1\right ] & \left [2,2.01\right ]\\\hline \text {Average velocity} & \class{xm-answer}{\cssId{xm-answer-6}{\phantom{\rule{7em}{1.5em}}}} & \class{xm-answer}{\cssId{xm-answer-7}{\phantom{\rule{7em}{1.5em}}}} & \class{xm-answer}{\cssId{xm-answer-8}{\phantom{\rule{7em}{1.5em}}}} & \class{xm-answer}{\cssId{xm-answer-9}{\phantom{\rule{7em}{1.5em}}}} & \class{xm-answer}{\cssId{xm-answer-10}{\phantom{\rule{7em}{1.5em}}}} \\\hline \end{array} \]
Um bom número inteiro para estimar a velocidade instantânea do objeto no instante \(t= 2\)
é \(\class{xm-answer}{\cssId{xm-answer-11}{\phantom{\rule{7em}{1.5em}}}}\)\((ft/s)\).
Usando limites, calcule a velocidade instantânea do objeto no instante \(t=2\).
\[ v(2) = \lim _{t\to 2}\frac {s(t)-s(2)}{ \class{xm-answer}{\cssId{xm-answer-12}{\phantom{\rule{7em}{1.5em}}}}} = \lim _{t\to 2}\frac { \class{xm-answer}{\cssId{xm-answer-13}{\phantom{\rule{7em}{1.5em}}}} -\class{xm-answer}{\cssId{xm-answer-14}{\phantom{\rule{7em}{1.5em}}}}}{\class{xm-answer}{\cssId{xm-answer-15}{\phantom{\rule{7em}{1.5em}}}}} = \lim _{t\to 2}\frac { t^2\Bigl (\class{xm-answer}{\cssId{xm-answer-16}{\phantom{\rule{7em}{1.5em}}}}\Bigr )} {\class{xm-answer}{\cssId{xm-answer-17}{\phantom{\rule{7em}{1.5em}}}}} = \lim _{t\to 2}\class{xm-answer}{\cssId{xm-answer-18}{\phantom{\rule{7em}{1.5em}}}}=\class{xm-answer}{\cssId{xm-answer-19}{\phantom{\rule{7em}{1.5em}}}}\text {pés/s.} \]
![[Picture]](/rendered/definitionOfTheDerivative/exercises/secantLineLimit1-8e2b38cac399383b7640b282a29f1b63.svg)
![[Picture]](/rendered/definitionOfTheDerivative/exercises/secantLineLimit1-0ec9971ff2cc032576a0744dad8561a8.svg)
![[Picture]](/rendered/definitionOfTheDerivative/exercises/secantLineLimit1-9d817bbfea6aace81649df13d9e34a03.svg)