Download AP Calculus AB/BC Unit 3 Assignment Sheet and Student

AP Calculus AB/BC Unit 3
Assignment Sheet and Student Objectives
Unit 3 – Chapter 2b (BC test date 9/22/15)
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Hmw 2.4a – pg. 137: 9-36(x3), 37, 38, 43, 44 (Chain rule)
Hmw 2.4b – pg. 137: 45-97 every other odd (Chain rule)
Hmw 2.4c – pp. 139-140: 109, 110, 113, 115, 120, 123, 131 (Chain rule)
Hmw 2.5a – pp. 146: 1-37 EOO (Implicit differentiation)
Hmw 2.5b – pp. 147-148: 41, 45, 48, 51, 53, 57, 59, 65, 73, 74, 77 (Implicit differentiation)
Hmw 2.6a – pp. 154: 1-4, 5 & 7 IN CLASS TOGETHER AFTER THE LECTURE (Related Rates)
Hmw 2.6b – pp. 154-155: 11 – 25 odd
Hmw 2.6c – pp. 155-157: 31, 33, 43, 44, 50, 53
Objectives – Students will be able to…
1.
2.
3.
4.
5.
Find derivatives using the Chain Rule and the General Power
Rule.
Simplify derivatives using algebra.
Use the Chain Rule to find the derivative of trigonometric
functions.
Distinguish between functions written in implicit form and
explicit form.
Use implicit differentiation to find the derivative of a
function.
6. Find related rates and use them to solve real-life problems.
Chain Rule
If y = f(g(x)) then
𝑑𝑦
𝑑𝑥
= 𝑓 ′ (𝑔(𝑥)) ∙ 𝑔′ (𝑥).
Sometimes teachers refer to the chain rule as the “peanut m&m” rule.
A function like this one… 𝑦 = (3𝑥 + 2)4 … is like a peanut m&m. The ( )4 is like the shell, and the 3x + 2
is like the peanut.
To find the derivative of this kind of function, find the derivative of the shell and multiply it to the
derivative of the peanut.
m&m (The function)
(3𝑥 + 2)4
𝑠𝑖𝑛3 (𝑥)
(3𝑥 2 )5
cos⁡(2𝑥 4 )
Derivative of the shell
4(3𝑥 + 2)3
3𝑠𝑖𝑛2 (𝑥)
5(3𝑥 2 )4
−sin⁡(2𝑥 4 )
Derivative of the Final answer: 𝑑𝑦 =
𝑑𝑥
peanut
3
4(3𝑥 + 2)3 ∙ 3 = 12(3𝑥 + 2)3
cos⁡(𝑥)
3𝑠𝑖𝑛2 (𝑥)cos⁡(𝑥)
6𝑥
5(3𝑥 2 )4 6𝑥 = 30𝑥(3𝑥 2 )4
3
8x
−8𝑥 3 𝑠𝑖𝑛(2𝑥 4 )
The chain can be applied multiple times, as many as needed to completely unravel the chain.
4
Ex: 𝑦 = 𝑠𝑖𝑛4 (3𝑥 + 2) think of this as 𝑦 = ⁡ (𝑠𝑖𝑛(3𝑥 + 4)) . So, ( )4 is the outer shell, sin( ) is the
“chocolate coating”, and 3x+4 is the peanut. To find the derivative you need to find d(shell)/dx,
d(chocolate)/dx, and d(peanut)/dx. Then multiply.
𝑑𝑦
= 4𝑠𝑖𝑛3 (3𝑥 + 2) ∙ cos(3𝑥 + 2) ∙ 3
𝑑𝑥
Use the Chain Rule to find the Derivative.
Note: #5 and #8 cannot be done because their radicands will always be negative, and even indexed
roots of negative radicands are not real numbers.
Implicit Differentiation
An Explicit equation is one where y is explicitly written as a function of x.
Ex: y = 3x2+ 2
So far you have only found derivatives for explicit functions.
An Implicit equation is one that, basically, isn’t solved explicitly for y.
Ex: x2 + y3 = 6
To find the derivative of an implicit equation. Take the derivative of both sides of the equation and treat
y as a sort of “unknown” function. Since it is “unknown” the only thing we can write for its derivative is
dy/dx.
Ex: Given x2 + y3 = 6 applying implicit differentiation gives 2𝑥 + 3𝑦 2
𝑑𝑦
𝑑𝑥
= 0.
Notice how y3 was treated like a chain rule problem. If we had (f(x))3, chain rule would make the
derivative 3(f(x))2f’(x). Here it is the same idea. The y3 is the shell and y is the peanut. The d(shell)/dx is
3y2. The d(peanut)/dx is dy/dx.
To finish the example, solve for dy/dx. So the derivative of x2 + y3 = 6 by implicit differentiation is…
𝑑𝑦 −2𝑥
=
𝑑𝑥 3𝑦 2
Related Rates
Chain rule and implicit differentiation work together to solve related rates problems. The textbook gives
the following tips for solving these problems.
1. Identify all given quantities and quantities to be determined. Make a sketch and label the
quantities. (This is a very important step. Make an actual list of these “known” and “to be
determined” things.)
2. Write an equation involving the variables whose rates of change either are given or are to be
determined. (Almost “step 1A”. You want to make a list of the dy/dt or dr/dt or whatever rates
you actually know and whatever rates you need to find. And then write an equation using the
variables and rates from step 1 and “1A”.)
3. Use the chain rule and implicit differentiation to find the derivatives of both sides of the
equation with respect to time.
4. Plug in all of the “knowns” into your derivative and solve for the unknown rate.