q = f(p) = 10300e^−0.34p
a) Find the number of products sold when the price of the product is $5. (Round your answer to the nearest whole number.)
Number of products sold:
b) Find a formula for the rate of change in the number of products sold when the price is p dollars.
f�Œ(p) =
c) What is the rate of change in the number of products sold when the price is $5? (Round your answer to one decimal place.)
Rate of Change:
by the time you get to calculus, you certainly should know how to plug in a value and evaluate a function! Where do you get stuck?
(a) q(5) = 10300*e^(-0.34*5) = 1882
(b) as you know, if
y = e^u
y' = e^u u'
And, as you know, the derivative is the rate of change.
So, q' = 10300*e^(-0.34p)(-.34) = -3502e^(-.34p)
(c) now plug in p=5 to get q'(5)
extra credit: what does it mean that q'(p) is negative?
satisfy but not enough
a) To find the number of products sold when the price is $5, we substitute p = 5 into the equation q = 10300e^(-0.34p):
q = 10300e^(-0.34*5)
≈ 10300e^(-1.7)
≈ 10300 * 0.1822
≈ 1875.9
So, the number of products sold when the price is $5 is approximately 1876.
b) To find the formula for the rate of change in the number of products sold when the price is p dollars, we need to find the derivative of the function q:
f'(p) = d(q)/d(p) = d(10300e^(-0.34p))/d(p)
Using the chain rule, the derivative of e^(ax) is ae^(ax), so:
f'(p) = 10300 * (-0.34) * (e^(-0.34p))
= -3502e^(-0.34p)
Therefore, the formula for the rate of change in the number of products sold when the price is p dollars is f�Œ(p) = -3502e^(-0.34p).
c) To find the rate of change in the number of products sold when the price is $5, we substitute p = 5 into the rate of change formula:
f�Œ(5) = -3502e^(-0.34*5)
≈ -3502e^(-1.7)
≈ -3502 * 0.1822
≈ -638.0 (rounded to one decimal place)
So, the rate of change in the number of products sold when the price is $5 is approximately -638.0.
a) To find the number of products sold when the price of the product is $5, we need to substitute p = 5 into the equation q = 10300e^(-0.34p).
Substituting p = 5 into the equation:
q = 10300e^(-0.34(5))
Calculating the exponent:
q = 10300e^(-0.34(5))
q = 10300e^(-1.7)
Calculating the value of q using a calculator:
q ≈ 10300 * 0.18268352405273525
q ≈ 1881.94
Rounding to the nearest whole number:
Number of products sold ≈ 1882
b) To find a formula for the rate of change in the number of products sold when the price is p dollars, we need to differentiate the equation q = 10300e^(-0.34p) with respect to p.
Differentiating the equation q = 10300e^(-0.34p):
f'(p) = -0.34 * 10300 * e^(-0.34p)
Simplifying the formula:
f�'(p) = -3502e^(-0.34p)
The formula for the rate of change in the number of products sold when the price is p dollars is f'(p) = -3502e^(-0.34p).
c) To find the rate of change in the number of products sold when the price is $5, we need to substitute p = 5 into the formula for the rate of change f'(p) = -3502e^(-0.34p).
Substituting p = 5 into the formula:
Rate of Change = -3502e^(-0.34(5))
Calculating the exponent:
Rate of Change = -3502e^(-0.34(5))
Rate of Change ≈ -3502e^(-1.7)
Calculating the value of the rate of change using a calculator:
Rate of Change ≈ -3502 * 0.18268352405273525
Rounding to one decimal place:
Rate of Change ≈ -639.4