Questions: Suppose customers in a hardware store are willing to buy N(p) boxes of nails at p dollars per box, as given by the following function. Complete parts (a) through (d). N(p)=80-3 p^2:(2 ≤ p ≤ 5) (Type an integer or a decimal.) (b) The instantaneous rate of change for a function f when x=a is given by which formula? A. lim h→0 (f(a-h))/h, provided the limit exists B. lim h→0 (f(a-h)+f(a))/h, provided the limit exists C. lim h→0 (f(a+h)-f(a))/h, provided the limit exists D. lim h→0 f(a)/h, provided the limit exists Find the instantaneous rate of change of demand when the price is 3. The instantaneous rate of change of demand when the price is 3 is

Suppose customers in a hardware store are willing to buy N(p) boxes of nails at p dollars per box, as given by the following function. Complete parts (a) through (d).
N(p)=80-3 p^2:(2 ≤ p ≤ 5)
(Type an integer or a decimal.)
(b) The instantaneous rate of change for a function f when x=a is given by which formula?
A. lim h→0 (f(a-h))/h, provided the limit exists
B. lim h→0 (f(a-h)+f(a))/h, provided the limit exists
C. lim h→0 (f(a+h)-f(a))/h, provided the limit exists
D. lim h→0 f(a)/h, provided the limit exists

Find the instantaneous rate of change of demand when the price is 3.
The instantaneous rate of change of demand when the price is 3 is

Transcript text: Suppose customers in a hardware store are willing to buy $N(p)$ boxes of nails at $p$ dollars per box, as given by the following function. Complete parts (a) through (d). \[ N(p)=80-3 p^{2}:(2 \leq p \leq 5) \] (Type an integer or a decimal.) (b) The instantaneous rate of change for a function f when $\mathrm{x}=\mathrm{a}$ is given by which formula? A. $\lim _{h \rightarrow 0} \frac{f(a-h)}{h}$, provided the limit exists B. $\lim _{h \rightarrow 0} \frac{f(a-h)+f(a)}{h}$, provided the limit exists C. $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$, provided the limit exists D. $\lim _{h \rightarrow 0} \frac{f(a)}{h}$, provided the limit exists Find the instantaneous rate of change of demand when the price is $\$ 3$. The instantaneous rate of change of demand when the price is $\$ 3$ is $\square$ $\square$ (Type an integer or a decimal.)

Solution

Solution Steps

To solve the given problem, we need to address the following parts:

(a) We need to find the instantaneous rate of change of the demand function $ N(p) = 80 - 3p^2 $ at a specific price point, $ p = 3 $. This involves finding the derivative of the function $ N(p) $ and evaluating it at $ p = 3 $.

(b) The question asks for the correct formula for the instantaneous rate of change, which is the definition of the derivative. The correct choice is option C: $\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}$.

Step 1: Define the Demand Function

The demand function is given by: \[ N(p) = 80 - 3p^2 \] where $ p $ is the price per box of nails.

Step 2: Differentiate the Demand Function

To find the instantaneous rate of change of demand, we need to differentiate the function $ N(p) $ with respect to $ p $: \[ N'(p) = \frac{d}{dp}(80 - 3p^2) = -6p \]

Step 3: Evaluate the Derivative at $ p = 3 $

Substitute $ p = 3 $ into the derivative to find the instantaneous rate of change at this price: \[ N'(3) = -6 \times 3 = -18 \]

Step 4: Identify the Correct Formula for Instantaneous Rate of Change

The instantaneous rate of change of a function $ f $ at $ x = a $ is given by the derivative, which is defined as: \[ \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \] Thus, the correct choice is C.

Final Answer

$\boxed{-18}$

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