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 $\lim\limits_{t\to x}\frac{\fe{f}{t}-\fe{f}{x}}{t-x}$.

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 $\lim\limits_{h\to0}\frac{\fe{\cos}{h}-1}{h}=0$. Use these limits to help you to establish the first derivative formula for $\fe{\sin}{x}$.

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 $\fe{s}{t}=150+60t-16t^2$.

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 ftmin).

8

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

9

$\fe{v}{t}$ is the velocity of the space shuttle mih 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).

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 $\fe{\fd{f}}{x}=\frac{10x^4+8x^2-10x+6}{25x^4+30x^2+9}$.

19

Determine the equation of the tangent line to $f$ at $1$.

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?