Solved on Feb 03, 2024

Find the demand function $D(p) = -2p^2 - 6p + 300$ and its rate of change w.r.t. price $p$ . Interpret the rate of change when $p = \$11$ .

STEP 1

Assumptions
1. The demand function is given by $D(p)=-2p^2-6p+300$ , where $p$ is the price in dollars.
2. We need to find the rate of change of demand with respect to price, which is the derivative of the demand function with respect to $p$ .
3. We also need to evaluate the derivative at $p = \$11$ to find the rate of change of demand at that price.

STEP 2

To find the rate of change of demand with respect to price, we need to differentiate the demand function $D(p)$ with respect to $p$ .
$\frac{dD}{dp} = \frac{d}{dp}(-2p^2 - 6p + 300)$

STEP 3

Apply the power rule of differentiation to each term of the function.
$\frac{dD}{dp} = -2 \cdot \frac{d}{dp}(p^2) - 6 \cdot \frac{d}{dp}(p) + \frac{d}{dp}(300)$

STEP 4

Differentiate each term.
$\frac{dD}{dp} = -2 \cdot (2p) - 6 \cdot (1) + 0$

STEP 5

Simplify the derivative.
$\frac{dD}{dp} = -4p - 6$
This is the rate of change of demand with respect to price.

STEP 6

Now we need to find the rate of change of demand when the price is $p = \$11$ . We do this by substituting $p = 11$ into the derivative we found.
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6$

STEP 7

Calculate the value of the derivative at $p = 11$ .
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6 = -44 - 6$

STEP 8

Simplify the expression to find the rate of change of demand at $p = \$11$ .
$\frac{dD}{dp}\Big|_{p=11} = -50$

STEP 9

Interpret the result. A rate of change of $-50$ means that for each dollar increase in price, the demand decreases by 50 units. Conversely, for each dollar decrease in price, the demand increases by 50 units.
a. The rate of change of demand with respect to price is $-4p - 6$ . b. The rate of change of demand when the price is $\$11$ is $-50$ , meaning that if the price increases by $1, the demand will decrease by 50 units.

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Find the demand function $D(p) = -2p^2 - 6p + 300$ and its rate of change w.r.t. price $p$ . Interpret the rate of change when $p = \$11$ .

STEP 1

STEP 2

To find the rate of change of demand with respect to price, we need to differentiate the demand function $D(p)$ with respect to $p$ .
$\frac{dD}{dp} = \frac{d}{dp}(-2p^2 - 6p + 300)$

STEP 3

Apply the power rule of differentiation to each term of the function.
$\frac{dD}{dp} = -2 \cdot \frac{d}{dp}(p^2) - 6 \cdot \frac{d}{dp}(p) + \frac{d}{dp}(300)$

STEP 4

Differentiate each term.
$\frac{dD}{dp} = -2 \cdot (2p) - 6 \cdot (1) + 0$

STEP 5

Simplify the derivative.
$\frac{dD}{dp} = -4p - 6$
This is the rate of change of demand with respect to price.

STEP 6

Now we need to find the rate of change of demand when the price is $p = \$11$ . We do this by substituting $p = 11$ into the derivative we found.
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6$

STEP 7

Calculate the value of the derivative at $p = 11$ .
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6 = -44 - 6$

STEP 8

Simplify the expression to find the rate of change of demand at $p = \$11$ .
$\frac{dD}{dp}\Big|_{p=11} = -50$

STEP 9

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Find the demand function D(p)=−2p2−6p+300D(p) = -2p^2 - 6p + 300D(p)=−2p2−6p+300 and its rate of change w.r.t. price ppp. Interpret the rate of change when p=$11p = \$11p=$11.

STEP 1

STEP 2

To find the rate of change of demand with respect to price, we need to differentiate the demand function D(p)D(p)D(p) with respect to ppp.dDdp=ddp(−2p2−6p+300)\frac{dD}{dp} = \frac{d}{dp}(-2p^2 - 6p + 300)dpdD​=dpd​(−2p2−6p+300)

STEP 3

Apply the power rule of differentiation to each term of the function.dDdp=−2⋅ddp(p2)−6⋅ddp(p)+ddp(300)\frac{dD}{dp} = -2 \cdot \frac{d}{dp}(p^2) - 6 \cdot \frac{d}{dp}(p) + \frac{d}{dp}(300)dpdD​=−2⋅dpd​(p2)−6⋅dpd​(p)+dpd​(300)

STEP 4

Differentiate each term.dDdp=−2⋅(2p)−6⋅(1)+0\frac{dD}{dp} = -2 \cdot (2p) - 6 \cdot (1) + 0dpdD​=−2⋅(2p)−6⋅(1)+0

STEP 5

Simplify the derivative.dDdp=−4p−6\frac{dD}{dp} = -4p - 6dpdD​=−4p−6This is the rate of change of demand with respect to price.

STEP 6

Now we need to find the rate of change of demand when the price is p=$11p = \$11p=$11. We do this by substituting p=11p = 11p=11 into the derivative we found.dDdp∣p=11=−4(11)−6\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6dpdD​∣∣​p=11​=−4(11)−6

STEP 7

Calculate the value of the derivative at p=11p = 11p=11.dDdp∣p=11=−4(11)−6=−44−6\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6 = -44 - 6dpdD​∣∣​p=11​=−4(11)−6=−44−6

STEP 8

Simplify the expression to find the rate of change of demand at p=$11p = \$11p=$11.dDdp∣p=11=−50\frac{dD}{dp}\Big|_{p=11} = -50dpdD​∣∣​p=11​=−50

STEP 9

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Find the demand function $D(p) = -2p^2 - 6p + 300$ and its rate of change w.r.t. price $p$ . Interpret the rate of change when $p = \$11$ .

To find the rate of change of demand with respect to price, we need to differentiate the demand function $D(p)$ with respect to $p$ .
$\frac{dD}{dp} = \frac{d}{dp}(-2p^2 - 6p + 300)$

Apply the power rule of differentiation to each term of the function.
$\frac{dD}{dp} = -2 \cdot \frac{d}{dp}(p^2) - 6 \cdot \frac{d}{dp}(p) + \frac{d}{dp}(300)$

Differentiate each term.
$\frac{dD}{dp} = -2 \cdot (2p) - 6 \cdot (1) + 0$

Simplify the derivative.
$\frac{dD}{dp} = -4p - 6$
This is the rate of change of demand with respect to price.

Now we need to find the rate of change of demand when the price is $p = \$11$ . We do this by substituting $p = 11$ into the derivative we found.
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6$

Calculate the value of the derivative at $p = 11$ .
$\frac{dD}{dp}\Big|_{p=11} = -4(11) - 6 = -44 - 6$

Simplify the expression to find the rate of change of demand at $p = \$11$ .
$\frac{dD}{dp}\Big|_{p=11} = -50$