Transcript 3.6 Implicit Differentiation and Related Rates

Section 3.6
1. Implicitly differentiate to find dy/dx.
y3 – x2 = 4
2. Implicitly differentiate to find dy/dx.
y 4 – x 3 = 2x
3. Implicitly differentiate to find dy/dx.
x 2y = 8
4. Implicitly differentiate to find dy/dx.
x(y – 1) 2 = 6
5. Implicitly differentiate to find dy/dx.
1 1
 2
x y
6. Implicitly differentiate to find dy/dx and evaluate it at the given point.
y 2 – x 3 = 1 at x = 2, y = 3.
7. Implicitly differentiate to find dy/dx and evaluate it at the given point.
x 2 y + y 2x = 0 at x = - 2, y = 2.
8. For the following demand equation Implicitly differentiate to find dp/dx.
p 2 + p + 2x = 100
9. For the following demand equation Implicitly differentiate to find dp/dx.
xp 3 = 36
10. BUSINESS: Demand Equation – A company’s demand equation is
x = (2000 – p 2 ) ½ , where p is the price in dollars. Find dp/dx when
p = 50 and interpret your answer.
Implicitly differentiate the given function.
11. BUSINESS: Sales – The number of MP3 music players that a store
will sell and their price p (in dollars) are related by the equation
2x 2 = 15,000 – p 2 . Find dx/dp when p = 100 and interpret
your answer.
12. BUSINESS: Work Hours - A management consultant estimates that the number h of
hours per day that employees will work and their daily pay pf p dollars are related by the
equation 60h5 + 2,000,000 = p3. Find dh/dp at p = 200 and interpret your answer.
13. GENERAL” Snowball - A large snowball is melting so that its radius is decreasing at a
rate of 2 inches per hour. How fast is the volume decreasing at the moment when the
radius is 3 inches away? [The volume if a sphere of a radius r is V = 4/3 (pie) r3 ]
14. Business: Revenue - A company's revenue from selling x units of an item is given as R
= 100x – x2 dollars. If sales are increasing at the rate of 80 per day, find how rapidly
revenue is growing (in dollars per day) when 400 units have been sold.
R = 1000x – x2
R’ = 1000x’ – 2xx’
= x’(1000 – 2x)
The quantity sold, x, is increasing at 80 per day.
i.e., x’ = 80 when x = 400.
R’ = 80[1000 – 2(400)] = 16,000
Therefore the company’s revenue is increasing at $ 16,000 per day.
15. Related Rates: Speeding - A traffic patrol helicopter is stationary a quarter of a mile
directly above a highway, as shown below. It’s radar detects a car whose line-of-sight
distance from the helicopter is half a mile and is increasing at the rate of 57 mph. Is the
car exceeding the highway’s speed limit of 60 mph?
0.25
16. Business: Sales - The number x of printers cartridges that a store will sell per week
and their price p (in dollars) are related by the equation x 2 = 4500 – 5p 2. If the price is
falling at the rate of $1 per week, find how the sales will change if the current price is
$20.