THIS QUESTION REQUIRES A COMPUTED ANSWER, NOT AN EXPRESSION. Please help :(

Question

star233 · Accepted Answer

To compute the Taylor series approximation for the given function, we need to find the first and second derivatives of the function, evaluate them at

x_{0}

, and plug the values into the Taylor series formula.
Let's start by finding the first derivative of

f (x) = - 4 x^{2} + 7 x + 12

. We can use the power rule for differentiation:

f^{'} (x) = \frac{d}{d x} (- 4 x^{2} + 7 x + 12)

To differentiate each term, we bring down the exponent as a coefficient and decrease the exponent by 1:

f^{'} (x) = - 8 x + 7

Next, let's find the second derivative of f(x) by differentiating the first derivative:

f^{″} (x) = \frac{d}{d x} (- 8 x + 7)

Again, differentiating each term gives us:

f^{″} (x) = - 8

Now, we have the necessary derivatives to compute the Taylor series approximation for

t r i a n g l e f

, denoted as Δf(x₀). The Taylor series formula for Δf is given by:

Δ f = f^{'} (x ₀) \cdot Δ x + \frac{f^{″} (x ₀)}{2} \cdot (Δ x)^{2}

Plugging in the values for x₀ = 8 and the derivatives, we get:

Δ f = f^{'} (8) \cdot Δ x + \frac{f^{″} (8)}{2} \cdot (Δ x)^{2}

Now, we need to evaluate the derivatives at x₀ = 8:

f^{'} (8) = - 8 (8) + 7 = - 57

f^{″} (8) = - 8

Substituting these values into the formula, we have:

Δ f = (- 57) \cdot Δ x + \frac{(- 8)}{2} \cdot (Δ x)^{2}

Simplifying further:

Δ f = - 57 Δ x - 4 (Δ x)^{2}

This is the Taylor series approximation to

t r i a n g l e f

for

x_{0} = 8

, given the function

f (x) = - 4 x^{2} + 7 x + 12

.

THIS QUESTION REQUIRES A COMPUTED ANSWER, NOT AN