Questions: Estimate h'(1.5) using the table above. h'(1.5) ≈

Transcript text: Estimate $h^{\prime}(1.5)$ using the table above. \[ h^{\prime}(1.5) \approx \]

Solution

Solution Steps

To estimate $ h^{\prime}(1.5) $, we can use the concept of numerical differentiation, specifically the symmetric difference quotient. This involves using the values of $ h(q) $ at points around 1.5, which are 1 and 2 in this case. The formula for the symmetric difference quotient is:

\[ h^{\prime}(1.5) \approx \frac{h(2) - h(1)}{2 - 1} \]

Step 1: Identify the Relevant Data Points

To estimate $ h^{\prime}(1.5) $, we need the values of $ h(q) $ at points around $ q = 1.5 $. From the given data, these points are $ q = 1 $ and $ q = 2 $ with corresponding $ h(q) $ values of 31 and 141, respectively.

Step 2: Apply the Symmetric Difference Quotient

The symmetric difference quotient formula for estimating the derivative at a point is:

\[ h^{\prime}(1.5) \approx \frac{h(2) - h(1)}{2 - 1} \]

Substituting the values from the data:

\[ h^{\prime}(1.5) \approx \frac{141 - 31}{2 - 1} = \frac{110}{1} = 110.0 \]