3.6: Derivatives as Rates of Change
- Page ID
- 144210
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- The position of a drone flying along a straight line is given by the function \(x(t) = 2t^3 - 6t\), where \(t\) is in seconds and \(x(t)\) is in meters.
- What is the initial velocity of the drone?
- What is the drone's velocity after \(t = 1\) second?
- What is the initial acceleration of the drone?
- What is the drone's acceleration after \(t = 1\) second?
- What is the initial velocity of the drone?
- A ball is dropped from the top of a building 256 feet tall. The ball's height in feet above the ground \(t\) seconds after it is released is given by \(y(t) = -16t^2 + 256\) .
- How long does it take for the ball to hit the ground?
- What is the velocity of the ball at the moment when it is released?
- What is the velocity of the ball when it hits the ground?
- What is the acceleration of the ball at the moment when it is released?
- What is the acceleration of the ball when it hits the ground?
- How long does it take for the ball to hit the ground?
- The position in miles of a freight train after \(t\) hours is given by the function \(s(t) = t^3 - 4t\), where east is the positive direction.
- Determine the direction and speed of the train when \(s(t) = 0\).
- Determine the direction and speed of the train when \(a(t) = 0\).
- Determine the direction and speed of the train when \(s(t) = 0\).
- A small town in Ohio commissioned an actuarial firm to conduct a study that modeled the rate of change of the town's population. The study found that the town's population can be modeled by the function \(P(t) = -\dfrac{1}{3}t^3 + 64t + 3000\), where \(t\) is measured in years.
- Find \(P'(1)\), \(P'(2)\), \(P'(3)\), and \(P'(4)\). Interpret what the results mean for the town's population.
- Find \(P''(1)\), \(P''(2)\), \(P''(3)\), and \(P''(4)\). Interpret what the results mean for the town's population.
- Find \(P'(1)\), \(P'(2)\), \(P'(3)\), and \(P'(4)\). Interpret what the results mean for the town's population.
- A culture of bacteria grows in number according to the function \(N(t) = 3000\left(1 + \dfrac{4t}{t^2 + 100}\right)\), where \(t\) is measured in hours.
- Find the rate of change of the number of bacteria, \(N'(t)\).
- Using a calculator, find \(N'(0)\), \(N'(10)\), \(N'(20)\), and \(N'(30)\). Interpret what the results imply about the bacteria population.
- Find \(N''(t)\).
- Using a calculator, find \(N''(0)\), \(N''(10)\), \(N''(20)\), and \(N''(30)\). Interpret what the results imply about the bacteria population.
- Find the rate of change of the number of bacteria, \(N'(t)\).
- The cost in dollars for a company to manufacture \(x\) food processors is given by \(C(x) = 200 + \dfrac{7}{x} + \dfrac{x}{27}\).
- Find the marginal cost of manufacturing 12 food processors. Using your result, estimate the cost of producing the thirteenth food processor.
- Find the actual cost of manufacturing the thirteenth food processor.
- Find the marginal cost of manufacturing 12 food processors. Using your result, estimate the cost of producing the thirteenth food processor.
- The total revenue (in dollars) that a company makes for manufacturing and selling \(x\) digital clock radios is \(R(x) = 10x - 0.001x^2\).
- Find the marginal revenue function.
- Calculate the marginal revenue for \(x = 2000\) and \(x = 5000\). Interpret your results.
- Find the marginal revenue function.