Questions: The daily demand, q, for cupcakes is a function of the price p, where q=f(p)=210-36p. a. Find and interpret f(1). b. Find and interpret f'(1). a. f(1)= Interpret f(1)=174 in economic terms. When cupcakes sell for each, there will be sold that day. b. f'(1)= Interpret f'(1)=-36 in economic terms. Lowering the price from to would cause a(n) in sales of cupcakes.

The daily demand, q, for cupcakes is a function of the price p, where q=f(p)=210-36p. a. Find and interpret f(1). b. Find and interpret f'(1). a. f(1)= Interpret f(1)=174 in economic terms. When cupcakes sell for each, there will be sold that day. b. f'(1)= Interpret f'(1)=-36 in economic terms. Lowering the price from to would cause a(n) in sales of cupcakes.
Transcript text: The daily demand, $q$, for cupcakes is a function of the price $p$, where $q=f(p)=210-36 p$. a. Find and interpret $f(1)$. b. Find and interpret $f^{\prime}(1)$. a. $f(1)=$ $\square$ Interpret $\mathrm{f}(1)=174$ in economic terms. When cupcakes sell for \$ $\square$ each, there will be $\square$ sold that day. b. $f^{\prime}(1)=$ $\square$ Interpret $f^{\prime}(1)=-36$ in economic terms. Lowering the price from $\$$ $\square$ to $\$$ $\square$ would cause a(n) $\square$ in sales of $\square$ cupcakes.
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Solution

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Solution Steps

Step 1: Calculate \( f(1) \)

The function \( f(p) = 210 - 36p \) represents the daily demand for cupcakes as a function of price \( p \). To find \( f(1) \), substitute \( p = 1 \) into the function: \[ f(1) = 210 - 36(1) = 210 - 36 = 174. \] Thus, \( f(1) = 174 \).

Step 2: Interpret \( f(1) = 174 \) in economic terms

The value \( f(1) = 174 \) means that when the price of cupcakes is \$1 each, the daily demand for cupcakes is 174.

Step 3: Calculate \( f^{\prime}(1) \)

The derivative of \( f(p) \) with respect to \( p \) is: \[ f^{\prime}(p) = \frac{d}{dp}(210 - 36p) = -36. \] Thus, \( f^{\prime}(1) = -36 \).

Step 4: Interpret \( f^{\prime}(1) = -36 \) in economic terms

The derivative \( f^{\prime}(1) = -36 \) represents the rate of change of demand with respect to price. Specifically, it means that for every \$1 decrease in price, the demand for cupcakes increases by 36.

Final Answer

a. \( f(1) = \boxed{174} \)

Interpretation: When cupcakes sell for \$1 each, there will be 174 sold that day.

b. \( f^{\prime}(1) = \boxed{-36} \)

Interpretation: Lowering the price from \$1 to \$0 would cause an increase in sales of 36 cupcakes.

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