Apply two decimal place rounding where applicable.The profit function of a certain firm isf(x) = -2x² + 396x - 400,where x denotes units of product produced, and f(x) is in Rands.Using thea) two-point forward difference formula (obtained from Taylor's Theorem), thefirm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.production x) for x = 20 and h = 1 is: b) two-point backward difference formula (obtained from Taylor's Theorem), thefirm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.production x) for xo = 20 and h = 1 is:f'(x₁) =c) two-point central difference formula (obtained from Taylor's Theorem), thefirm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.production x) for xo = 20 and h = 1 is:f'(xo) =d) three-point central difference formula (obtained from Taylor's Theorem) withXo = 20 and h = 1 yields:f" (xo) =

Answered: Apply two decimal place rounding where…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Apply two decimal place rounding where applicable.
The profit function of a certain firm is
f(x) = -2x² + 396x - 400,
where x denotes units of product produced, and f(x) is in Rands.
Using the
a) two-point forward difference formula (obtained from Taylor's Theorem), the
firm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.
production x) for x = 20 and h = 1 is:

b) two-point backward difference formula (obtained from Taylor's Theorem), the
firm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.
production x) for xo = 20 and h = 1 is:
f'(x₁) =
c) two-point central difference formula (obtained from Taylor's Theorem), the
firm's marginal profit (i.e., the instantaneous rate of change of its profit w.r.t.
production x) for xo = 20 and h = 1 is:
f'(xo) =
d) three-point central difference formula (obtained from Taylor's Theorem) with
Xo = 20 and h = 1 yields:
f" (xo) =

Expert Solution

Step 1

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What is Two Point Finite Difference Approximation:

Two point finite difference approximation is a tool to approximate the derivative of a function at a certain point. There are three rules. These are forward difference, backward difference and central difference. These rules are determined from Taylor's approximation theorem.

Given:

Given profit function is

$f (x) = - 2 x^{2} + 396 x - 400$

Here, $x$ represents units of product produces and $f (x)$ is in Rands.

Given,

$\begin{array}{rcl} x_{0} & = & 20 \\ h & = & 1 \end{array}$

To Determine:

We determine $f' (20)$ using two point forward, backward and central difference approximation.