Question

1. the table shows the margin of error in degrees for tennis serves hit at 100 mph with various amounts of topspin (in units of revolutions per second).

topspin (rps) (x) 20 40 60 80 100
margin of error f(x) 1.8 2.4 3.1 3.9 4.6
a.) estimate the derivative of f at x = 70 rps. label correctly.
b.) estimate the derivative of f at x = 40 rps. label correctly.

Explanation:

Step1: Recall the formula for the derivative approximation

The derivative of a function $y = f(x)$ at a point $x$ can be approximated using the difference - quotient formula: $f^{\prime}(x)\approx\frac{f(x + h)-f(x - h)}{2h}$.

Step2: Estimate the derivative at $x = 70$ rps

For $x = 70$ rps, we use the points $x_1 = 60$ and $x_2=80$. Here, $h = 10$.
$f(60)=3.1$, $f(80)=3.9$.
$f^{\prime}(70)\approx\frac{f(80)-f(60)}{80 - 60}=\frac{3.9 - 3.1}{20}=\frac{0.8}{20}=0.04$ degrees per rps.

Step3: Estimate the derivative at $x = 40$ rps

For $x = 40$ rps, we use the points $x_1 = 20$ and $x_2 = 60$. Here, $h = 20$.
$f(20)=1.8$, $f(60)=3.1$.
$f^{\prime}(40)\approx\frac{f(60)-f(20)}{60 - 20}=\frac{3.1 - 1.8}{40}=\frac{1.3}{40}=0.0325$ degrees per rps.

Answer:

a. $f^{\prime}(70)\approx0.04$ degrees per rps
b. $f^{\prime}(40)\approx0.0325$ degrees per rps