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 \(\lim\limits_{t\to x}\frac{\fe{f}{t}-\fe{f}{x}}{t-x}\text{.}\)

# Activity3.5Supplement¶ permalink

# Subsection3.5.1Exercises

##### 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 \(\fe{\sin}{x}\text{.}\)

HintSuppose 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.