Q. 0 - Calculus

Question

Problem Zero: Read the section and make your own summary of the material.

Step-by-Step Solution

Answer

$$\begin{gathered} 1. \mathit{F}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathit{l}\mathit{y}\mathbf{ }\mathit{D}\mathit{e}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Suppose f and f\text{'} are both differentiable on an interval I. \\ \left(a\right) f is concave up on I if f\text{'} is increa\sin g on I. \\ \left(b\right) f is concave down on I if f\text{'} is decrea\sin g on I. \end{gathered}$$

$$\begin{gathered} 2. \mathit{T}\mathit{h}\mathit{e}\mathbf{ }\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{ }\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{D}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\mathbf{ }\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Suppose both f and f\text{'} are differentiable on an interval I. \\ \left(a\right) If f\text{'}\text{'} is positive on I, then f is concave up on I. \\ \left(b\right) If f\text{'}\text{'} is negative on I, then f is concave down on I. \end{gathered}$$

$$\begin{gathered} 3. \mathit{T}\mathit{h}\mathit{e}\mathbf{ }\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{-}\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{T}\mathit{e}\mathit{s}\mathit{t}\mathbf{:} \\ Suppose x = c is the location of a critical point of a function f with f\text{'}\left(c\right) = 0, \\ and suppose both f and f\text{'} are differentiable and f\text{'}\text{'} is continuous on an interval \\ around x = c. \\ \left(a\right) If f\text{'}\text{'}\left(c\right) is positive, then f has a local minimum at x = c. \\ \left(b\right) If f\text{'}\text{'}\left(c\right) is negative, then f has a local maximum at x = c. \\ \left(c\right) If f\text{'}\text{'}\left(c\right) = 0, then this test says nothing about whether or not f has an extremum \\ at x = c. \end{gathered}$$

Step 1. Given Information.

Given is the text and next step contains summary of the text.

Step 2. Summary pf the text given.

$$\begin{gathered} 1. \mathit{F}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathit{l}\mathit{y}\mathbf{ }\mathit{D}\mathit{e}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{n}\mathit{g}\mathbf{ }\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Suppose f and f\text{'} are both differentiable on an interval I. \\ \left(a\right) f is concave up on I if f\text{'} is increa\sin g on I. \\ \left(b\right) f is concave down on I if f\text{'} is decrea\sin g on I. \end{gathered}$$

$$\begin{gathered} 2. \mathit{T}\mathit{h}\mathit{e}\mathbf{ }\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{ }\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{D}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathit{m}\mathit{i}\mathit{n}\mathit{e}\mathit{s}\mathbf{ }\mathit{C}\mathit{o}\mathit{n}\mathit{c}\mathit{a}\mathit{v}\mathit{i}\mathit{t}\mathit{y}\mathbf{:} \\ Suppose both f and f\text{'} are differentiable on an interval I. \\ \left(a\right) If f\text{'}\text{'} is positive on I, then f is concave up on I. \\ \left(b\right) If f\text{'}\text{'} is negative on I, then f is concave down on I. \end{gathered}$$

$$\begin{gathered} 3. \mathit{T}\mathit{h}\mathit{e}\mathbf{ }\mathit{S}\mathit{e}\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathbf{-}\mathit{D}\mathit{e}\mathit{r}\mathit{i}\mathit{v}\mathit{a}\mathit{t}\mathit{i}\mathit{v}\mathit{e}\mathbf{ }\mathit{T}\mathit{e}\mathit{s}\mathit{t}\mathbf{:} \\ Suppose x = c is the location of a critical point of a function f with f\text{'}\left(c\right) = 0, \\ and suppose both f and f\text{'} are differentiable and f\text{'}\text{'} is continuous on an interval \\ around x = c. \\ \left(a\right) If f\text{'}\text{'}\left(c\right) is positive, then f has a local minimum at x = c. \\ \left(b\right) If f\text{'}\text{'}\left(c\right) is negative, then f has a local maximum at x = c. \\ \left(c\right) If f\text{'}\text{'}\left(c\right) = 0, then this test says nothing about whether or not f has an extremum \\ at x = c. \end{gathered}$$