Question

this is section 3.1 problem 14: for y = f(x)=-\frac{2}{x}, x = 2, and \delta x=0.2: \delta y = 1/11 and f(x)\delta x 1/11 hint: follow example 2. round to three decimal places unless the exact answer has less decimal places. this is section 3.1 problem 16: for y = f(x)=\frac{3}{x^{2}}, x = 1, and \delta x=0.03: \delta y = and f(x)\delta x . round to three decimal places unless the exact answer has less decimal places. hint: follow example 2.

Explanation:

1. First, find the derivative of \(y = f(x)=-\frac{2}{x}\):

• Using the power - rule for differentiation, if \(y =-\frac{2}{x}=-2x^{-1}\), then \(y^\prime=f^\prime(x)=(-2)\times(-1)x^{-2}=\frac{2}{x^{2}}\).

1. Then, calculate \(f^\prime(x)\Delta x\):

• Given \(x = 2\) and \(\Delta x=0.2\).
• First, find \(f^\prime(2)\):
• Substitute \(x = 2\) into \(f^\prime(x)=\frac{2}{x^{2}}\), so \(f^\prime(2)=\frac{2}{2^{2}}=\frac{2}{4}=0.5\).
• Then, calculate \(f^\prime(x)\Delta x\):
• \(f^\prime(2)\Delta x=0.5\times0.2 = 0.1\).

1. Next, calculate \(\Delta y\):

• First, find \(y_1=f(x)\) when \(x = 2\): \(y_1=-\frac{2}{2}=-1\).
• Then, find \(x_2=x+\Delta x=2 + 0.2=2.2\).
• Next, find \(y_2=f(x_2)\) when \(x_2 = 2.2\): \(y_2=-\frac{2}{2.2}=-\frac{20}{22}=-\frac{10}{11}\approx - 0.909\).
• Then, \(\Delta y=y_2 - y_1=-\frac{10}{11}-(-1)=1-\frac{10}{11}=\frac{1}{11}\approx0.091\).

Step1: Find the derivative of \(y = f(x)\)

Using power - rule \((x^n)^\prime=nx^{n - 1}\), for \(y=-2x^{-1}\), \(y^\prime=\frac{2}{x^{2}}\).

Step2: Calculate \(f^\prime(x)\Delta x\) at \(x = 2\)

Substitute \(x = 2\) into \(f^\prime(x)\) to get \(f^\prime(2)=0.5\), then multiply by \(\Delta x = 0.2\).

Step3: Calculate \(\Delta y\)

Find \(y\) values at \(x\) and \(x+\Delta x\) and subtract.

Answer:

\(f^\prime(x)\Delta x = 0.1\), \(\Delta y\approx0.091\)