Skip to main content
\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Activity3.5Supplement

Subsection3.5.1Exercises

Find the first derivative formula for each of the following functions twice: first by evaluating \(\lim\limits_{h\to0}\frac{\fe{f}{x+h}-\fe{f}{x}}{h}\) and then by evaluating \(\lim\limits_{t\to x}\frac{\fe{f}{t}-\fe{f}{x}}{t-x}\text{.}\)

1

\(\fe{f}{x}=x^2\)

2

\(\fe{f}{x}=\sqrt{x}\)

3

\(\fe{f}{x}=7\)

4

It can be shown that \(\lim\limits_{h\to0}\frac{\fe{\sin}{h}}{h}=1\) and \(\lim\limits_{h\to0}\frac{\fe{\cos}{h}-1}{h}=0\text{.}\) Use these limits to help you to establish the first derivative formula for \(\fe{\sin}{x}\text{.}\)

Hint

Suppose that an object is tossed into the air in such a way that the elevation of the object (measured in ft) \(t\) seconds after the object was tossed is given by the function \(\fe{s}{t}=150+60t-16t^2\text{.}\)

5

Find the velocity function for this motion and use it to determine the velocity of the object \(4.1\) s into its motion.

6

Find the acceleration function for this motion and use that function to determine the acceleration of the object \(4.1\) s into its motion.

Determine the unit on the first derivative function for each of the following functions. Remember, we do not simplify derivative units in any way, shape, or form.

7

\(\fe{R}{p}\) is Carl's heart rate (beats per min) when he jogs at a rate of \(p\) (measured in ft⁄min).

8

\(\fe{F}{v}\) is the fuel consumption rate gal⁄mi of Hanh's pick-up when she drives it on level ground at a constant speed of \(v\) (measured in mi⁄h).

9

\(\fe{v}{t}\) is the velocity of the space shuttle mi⁄h where \(t\) is the amount of time that has passed since liftoff (measured in s).

10

\(\fe{h}{t}\) is the elevation of the space shuttle (mi) where \(t\) is the amount of time that has passed since lift-off (measured in s).

Referring to the functions in Exercises 3.5.1.7–10, write sentences that explain the meaning of each of the following function values.

11

\(\fe{R}{300}=84\)

12

\(\fe{\fd{R}}{300}=0.02\)

13

\(\fe{F}{50}=0.03\)

14

\(\fe{\fd{F}}{50}=-0.0006\)

15

\(\fe{v}{20}=266\)

16

\(\fe{\fd{v}}{20}=18.9\)

17

\(\fe{h}{20}=0.7\)

18

\(\fe{\fd{h}}{20}=0.074\)

It can be shown that the derivative formula for the function \(f\) defined by \(\fe{f}{x}=\frac{2x^3+2x+1}{3+5x^2}\) is \(\fe{\fd{f}}{x}=\frac{10x^4+8x^2-10x+6}{25x^4+30x^2+9}\text{.}\)

19

Determine the equation of the tangent line to \(f\) at \(1\text{.}\)

20

A graph of \(f\) is shown in Figure 3.5.1; axis scales have deliberately been omitted from the graph.

The graph shows that \(f\) quickly resembles a line. In a detailed sketch of \(f\) we would reflect this apparent linear behavior by adding a skew asymptote. What is the slope of this skew asymptote?

<<SVG image is unavailable, or your browser cannot render it>>

Figure3.5.1\(y=\fe{f}{x}=\frac{2x^3+2x+1}{3+5x^2}\)