ASU MAT210

Question 2

The table shows the number of cars sold per month at Lemon Cars where $t$ is in weeks and $S(t)$ is the number of cars sold. Find the average change in the number of cars sold between weeks 2 and 5.

The average rate of change in the number of cars sold was $5$ cars per week between weeks 2 and 5.

• Rationale: The average rate of change on the interval [t1​,t2​]=[2,5] is given by the formula:  ASU MAT210

  Average Rate of Change=5−2S(5)−S(2)​

  From the table, $S(5)=23$ and $S(2)=8$. Substituting these values:

  Average Rate of Change=5−223−8​=315​=5

Question 4

Let g(x)=4e−4x+4x3−ln(6x). Determine g′(x).

g′(x)=−16e−4x+12x2−x1​

• Rationale: Apply derivative rules term-by-term:

  1. For 4e−4x: dxd​[4e−4x]=4⋅(−4)e−4x=−16e−4x

  2. For 4x3: dxd​[4x3]=4⋅3x2=12x2

  3. For $\ln (6x)$: dxd​[ln(6x)]=6x1​⋅6=x1​ Combining these terms gives g′(x)=−16e−4x+12x2−x1​.

Question 5

Let f(x)=x7−2x5+8x3−4x−1.  ASU MAT210

• f′(x)=7x6−10x4+24x2−4

• f′′(x)=42x5−40x3+48x

• f′′(2)=1120

• Rationale:

  1. First derivative using the power rule:

     f′(x)=7x6−2(5)x4+8(3)x2−4=7x6−10x4+24x2−4

  2. Second derivative by taking the derivative of f′(x):

     f′′(x)=7(6)x5−10(4)x3+24(2)x=42x5−40x3+48x

  3. Evaluate f′′(2) by setting $x=2$:

     f′′(2)=42(2)5−40(2)3+48(2)=42(32)−40(8)+96=1344−320+96=1120

Question 6

Use the rules of derivatives to calculate the derivative of g(t)=5t​−3e4t+9t6.

g′(t)=51​t−4/5−12e4t+54t5

• Rationale: Rewrite 5t​ as t1/5. Then apply standard differentiation rules:

  1. dtd​[t1/5]=51​t1/5−1=51​t−4/5

  2. dtd​[−3e4t]=−3⋅4e4t=−12e4t

  3. dtd​[9t6]=9⋅6t5=54t5 Summing these terms yields g′(t)=51​t−4/5−12e4t+54t5.

Question 7

Given y=93v6+10 find dvdy​. ASU MAT210

dvdy​=2v5(3v6+10)−8/9

• Rationale: Rewrite the function with a fractional exponent: y=(3v6+10)1/9. Apply the general power rule (chain rule):

  dvdy​=91​(3v6+10)91​−1⋅dvd​[3v6+10]
  dvdy​=91​(3v6+10)−8/9⋅(18v5)=2v5(3v6+10)−8/9