Questions: Estimate the derivative of the given function at the indicated point by completing the entries of the given table for the function f(x). (Round your final answer to the nearest whole number. Values entered in the table should not be rounded.) f(x)=-9 x^2-2 x+4, a=-8 "h" "f(a+h)" "f(a+h)-f(a)" "(f(a+h)-f(a))/h" ------------ 0.1 -541.89 14.11 141.1 0.01 -554.5809 1.4191 141.91 0.001 0.141991 141.991 -0.1 -570.29 -14.29 142.9 -0.01 -557.4209 -1.4209 -0.001 -556.142009 -0.142009 142.09 f'(-8)=142

Solution Steps

To estimate the derivative of the function \( f(x) = -9x^2 - 2x + 4 \) at \( a = -8 \), we can use the definition of the derivative as the limit of the difference quotient. We will calculate the values of \( f(a+h) \) for small values of \( h \) and then use these to compute the difference quotient \( \frac{f(a+h) - f(a)}{h} \). Finally, we will round the result to the nearest whole number.

We are given the function \( f(x) = -9x^2 - 2x + 4 \) and we need to estimate its derivative at \( a = -8 \).

First, we calculate \( f(a) \) where \( a = -8 \): \[ f(-8) = -9(-8)^2 - 2(-8) + 4 = -9(64) + 16 + 4 = -576 + 16 + 4 = -556 \]

Next, we calculate \( f(a+h) \) for different values of \( h \): \[ \begin{aligned} f(-8 + 0.1) &= f(-7.9) = -9(-7.9)^2 - 2(-7.9) + 4 = -541.8900 \\ f(-8 + 0.01) &= f(-7.99) = -9(-7.99)^2 - 2(-7.99) + 4 = -554.5809 \\ f(-8 + 0.001) &= f(-7.999) = -9(-7.999)^2 - 2(-7.999) + 4 = -555.8580 \\ f(-8 - 0.1) &= f(-8.1) = -9(-8.1)^2 - 2(-8.1) + 4 = -570.2900 \\ f(-8 - 0.01) &= f(-8.01) = -9(-8.01)^2 - 2(-8.01) + 4 = -557.4209 \\ f(-8 - 0.001) &= f(-8.001) = -9(-8.001)^2 - 2(-8.001) + 4 = -556.1420 \\ \end{aligned} \]

We then calculate the difference quotient \( \frac{f(a+h) - f(a)}{h} \) for each \( h \): \[ \begin{aligned} \frac{f(-7.9) - f(-8)}{0.1} &= \frac{-541.8900 + 556}{0.1} = 141.1000 \\ \frac{f(-7.99) - f(-8)}{0.01} &= \frac{-554.5809 + 556}{0.01} = 141.9100 \\ \frac{f(-7.999) - f(-8)}{0.001} &= \frac{-555.8580 + 556}{0.001} = 141.9910 \\ \frac{f(-8.1) - f(-8)}{-0.1} &= \frac{-570.2900 + 556}{-0.1} = 142.9000 \\ \frac{f(-8.01) - f(-8)}{-0.01} &= \frac{-557.4209 + 556}{-0.01} = 142.0900 \\ \frac{f(-8.001) - f(-8)}{-0.001} &= \frac{-556.1420 + 556}{-0.001} = 142.0090 \\ \end{aligned} \]

To estimate the derivative, we average the difference quotients for the smallest positive and negative \( h \): \[ \frac{141.9910 + 142.0090}{2} = 142.0000 \] Rounding to the nearest whole number, we get: \[ f'(-8) \approx 142 \]

Final Answer