Question

estimating slope in exercises 5 and 6, use the graph to estimate the slope of the tangent line to ( y = x^n ) at the point ( (1, 1) ). verify your answer analytically. to print an enlarged copy of the graph, go to mathgraphs.com. 5. (a) ( y = x^{1/2} ) (b) ( y = x^3 ) graphs for 5(a) and 5(b) 6. (a) ( y = x^{-1/2} ) (b) ( y = x^{-1} ) graphs for 6(a) and 6(b)

Explanation:

5(a) \( y = x^{1/2} \)

Step1: Recall the power rule for differentiation

The power rule states that if \( y = x^n \), then \( y' = nx^{n - 1} \). For \( y = x^{1/2} \), we apply this rule.

Step2: Differentiate the function

Using the power rule with \( n=\frac{1}{2} \), we get \( y'=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}} \).

Step3: Evaluate the derivative at \( x = 1 \)

Substitute \( x = 1 \) into the derivative: \( y'(1)=\frac{1}{2\sqrt{1}}=\frac{1}{2} \).

Step1: Recall the power rule for differentiation

For \( y = x^n \), \( y' = nx^{n - 1} \). Here \( n = 3 \).

Step2: Differentiate the function

Applying the power rule, \( y' = 3x^{3 - 1}=3x^{2} \).

Step3: Evaluate the derivative at \( x = 1 \)

Substitute \( x = 1 \) into the derivative: \( y'(1)=3(1)^{2}=3 \).

Step1: Recall the power rule for differentiation

For \( y = x^n \), \( y' = nx^{n - 1} \). Here \( n=-\frac{1}{2} \).

Step2: Differentiate the function

Applying the power rule, \( y'=-\frac{1}{2}x^{-\frac{1}{2}-1}=-\frac{1}{2}x^{-\frac{3}{2}}=-\frac{1}{2x^{\frac{3}{2}}} \).

Step3: Evaluate the derivative at \( x = 1 \)

Substitute \( x = 1 \) into the derivative: \( y'(1)=-\frac{1}{2(1)^{\frac{3}{2}}}=-\frac{1}{2} \).

Answer:

The slope of the tangent line at \( (1,1) \) is \( \frac{1}{2} \).

5(b) \( y = x^{3} \)