4: Appropriate Applications
- Page ID
- 116583
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A rocket launch involves two related quantities that change over time. Being able to solve this type of problem is just one application of derivatives introduced in this chapter. We also look at how derivatives are used to find maximum and minimum values of functions. As a result, we will be able to solve applied optimization problems, such as maximizing revenue and minimizing surface area. In addition, we examine how derivatives are used to evaluate complicated limits, to approximate roots of functions, and to provide accurate graphs of functions.
- 4.1: Graphing Using Calculus - Maxima and Minima
- This section covers how to use calculus to graph functions by finding critical points, maxima, and minima. It explains how the first and second derivative tests help identify increasing or decreasing intervals and concavity, which are key to determining the shape of the graph. The section also demonstrates how to apply these concepts to real-world problems, optimizing functions to find their maximum or minimum values.
- 4.2: A Theoretical Interlude - The Mean Value Theorem
- This section introduces the Mean Value Theorem (MVT), which states that for a continuous and differentiable function, there exists at least one point on the interval where the tangent is parallel to the secant line connecting the endpoints. The section explains the significance of this theorem in calculus, its geometric interpretation, and provides examples to illustrate how to apply the MVT to solve problems.
- 4.3: Graphing Using Calculus - Shaping the Curve
- This section covers techniques for graphing functions by analyzing their shapes using calculus. It explains how the first and second derivatives indicate increasing/decreasing intervals, concavity, and inflection points. By applying these concepts, you can accurately sketch the function’s curve and identify key features such as peaks, valleys, and changes in direction.
- 4.4: An Interlude for Limits - L’Hospital’s Rule and Indeterminate Forms
- This section introduces L'Hôpital's Rule, a technique for evaluating limits that result in indeterminate forms such as \(0/0\) or \(\infty/\infty\). It explains how to apply the rule by differentiating the numerator and denominator until a determinate form is reached. The section also covers various indeterminate forms and provides examples to illustrate the use of L'Hôpital's Rule in solving complex limits.
- 4.5: Graphing Using Calculus - Putting it Altogether
- This section combines various calculus techniques for graphing functions comprehensively. It reviews how to use critical points, intervals of increase and decrease, concavity, inflection points, asymptotes, and end behavior to create accurate sketches of complex functions. By integrating these tools, you can analyze and fully understand the shape and behavior of graphs, applying both derivative and limit concepts to visualize functions effectively.
- 4.6: Optimization
- This section covers optimization, using calculus to find maximum or minimum values of functions in real-world applications. It explains setting up equations based on given constraints, finding critical points, and applying the first and second derivative tests to identify optimal solutions. Examples include problems in geometry, economics, and physics, illustrating practical uses of optimization.
- 4.7: Newton’s Method
- This section covers Newton's Method, a technique for approximating roots of a function. Starting from an initial guess, the method iteratively applies the formula \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) to get closer to the actual root. Examples illustrate how to use Newton's Method for finding solutions and demonstrate its effectiveness, especially when analytical solutions are challenging.