6.4: Summary of Chapter 6.

The thrust of this chapter is to expand your ability to compute derivatives of functions. We have now introduced all of the combination derivative formulas that you will need. Together with the Primary derivative formulas already introduced and two others to be presented in the next chapter, Chapter 7, Derivatives of the Trigonometric Functions, you will be able to compute the derivatives of all of the functions you will meet in ordinary work. The basic formulas that you need are shown below. You need to be able to use them forward and backward. That is, given a function, find its derivative, and given a derivative of a function, identify the function, or several such functions that have that same derivative. The backward process is crucial to the application of the Fundamental Theorem of Calculus, introduced in Chapter 10

The complete list of derivative formulas that you need is:

Definition: Term

Primary Formulas
\[\begin{aligned}
{[C]^{\prime} } &=0 & {\left[t^{n}\right]^{\prime} } &=n t^{n-1} \\
{\left[e^{t}\right]^{\prime} } &=e^{t} & {[\ln t]^{\prime} } &=\frac{1}{t} \\
{[\sin t]^{\prime} } &=\cos t & {[\cos t]^{\prime} } &=-\sin t
\end{aligned}\]
Combination formulas
\[\begin{aligned}
{[u+v]^{\prime} } &=[u]^{\prime}+[v]^{\prime} & {[C u]^{\prime} } &=C[u]^{\prime} \\
{[u v]^{\prime} } &=[u]^{\prime} v+u[v]^{\prime} & {\left[\frac{u}{v}\right]^{\prime} } &=\frac{v[u]^{\prime}-u[v]^{\prime}}{u^{2}} \\
{[G(u)]^{\prime} } &=G^{\prime}(u) u^{\prime} &
\end{aligned}\]

Chapter Exercise 6.4.1 Differentiate

1. \(P(t)=t^{4}+e^{2 t}\)
2. \(P(t)=t^{4} e^{2 t}\)
3. \(P(t)=\frac{t^{4}}{e^{2 t}}\)
4. \(P(t)=\left(e^{2 t}\right)^{4}\)
5. \(P(t)=e^{2 t^{4}}\)
6. \(P(t)=\left(t^{2}+1\right)^{4}(5 t+1)^{7}\)
7. \(P(t)=(\ln t)^{3}\)
8. \(P(t)=\left(e^{3 t} \ln 2 t\right)^{4}\)
9. \(P(t)=\frac{\ln t}{t}\)
10. \(P(t)=t \ln t-t\)
11. \(P(t)=\frac{1}{\ln t}\)
12. \(P(t)=e^{(\ln t)}\)
13. \(P(t)=\frac{5 t^{2}-2 t-7}{t^{2}+1}\)
14. \(P(t)=\frac{(t+2)^{2}}{t^{2}+2}\)

Chapter Exercise 6.4.2 Data from another of the ice ball experiments (see Exercise 6.2.6) are shown in Table 6.4.1.

1. Find a number \(A\) so that \(W(t) = 3200(1 − t/A) ^3\) is close to the data.
2. Find a number \(B\) so that \(W(t) = 3100(1 − t/B) ^2\) is close to the data.
3. Which of the two functions is closer to the data?

Close to the data may be defined in at least two ways. For several values of \(A\), compute

\[\begin{aligned}
S 1=&\left|w_{1}-3200\left(1-t_{1} / A\right)^{3}\right|+\\
&\left|w_{2}-3200\left(1-t_{2} / A\right)^{3}\right|+\cdots+\left|w_{21}-3200\left(1-t_{21} / A\right)^{3}\right|\\
S 2=&\left(w_{1}-3200\left(1-t_{1} / A\right)^{3}\right)^{2}+\\
&\left(w_{2}-3200\left(1-t_{2} / A\right)^{3}\right)^{2}+\cdots+\left(w_{21}-3200\left(1-t_{21} / A\right)^{3}\right)^{2}\\
\end{aligned}\]

and select the values, A1 and A2, of A for which S1 and S2, respectively, are the smallest. Then define \(W 1(t)=3200(1-t / A 1)^{3}\) and \(W 2(t)=3200(1-t / A 2)^{3}\). MATLAB code to do this follows.

Discuss the difference between S1 and S2.

Alter the code to do part b, and then answer part c.

Code \(\PageIndex{1}\) (MATLAB):

close all;clc;clear

t= [0:4:80];

w=[3085 2855 2591 2227 2085 1855 1645 1436 1245 1097 ...

      908 763 534 513 407 316 216 164 110 88 34];

AA = [80:1:120];

 

for i = 1:length(AA)

sum1(i) = 0.0; sum2(i) = 0.0;

for k = 1:21

sum1(i) = sum1(i)+ abs((w(k)-3200*(1-t(k)/AA(i))^3));

sum2(i) = sum2(i)+(w(k)-3200*(1-t(k)/AA(i))^3)^2;

end

end

 

[S1 I1] = min(sum1); A1=AA(I1)

[S2 I2] = min(sum2); A2=AA(I2)

 

W1=3200*(1-t/A1).^3; W2=3200*(1-t/A2).^3;

 

plot(t,w,’x’,t,W1,’o’,t,W2,’+’,’linewidth’,2);

 

Table for Chapter Exercise 6.4.1 Weight of an ice ball following immersion in \(8 ^{\circ} C\) water.

Time m	0	4	8	12	16	20	24	28	32	36
Wt g	3085	2855	2591	2337	2085	1855	1645	1436	1245	1097
Time m	40	44	48	52	56	60	64	68	72	76	80
Wt g	908	763	634	513	407	316	216	164	110	66	34