Download Lecture 4 - University of Chicago Math

Lecture 4
Review of Math 131
with Trig Functions
Math 13200
Trigonometry Derivative Rules
Recall from last time:
Dx sin x = cos x
Dx cos x = − sin x
Dx tan x = sec2 x
Dx cot x = − csc2 x
Dx sec x = sec x · tan x
Dx csc x = − csc x · cot x
Today, we will do a brief review of last quarter, but focusing on trig functions.
Derivative Rules (with trig functions)
For the following, let f, g be differentiable functions of x, k any constant, and r a rational number.
Rules from last quarter:
Constant
Dx k = 0
Identity
Dx x = 1
Constant Multiple
Dx (k · f ) = k · Dx f
Sum
Dx (f + g) = Dx f + Dx g
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Product
Dx (f · g) = f · Dx g + g · Dx f
Quotient
f
g · Dx f − f · Dx g
Dx
=
g
g2
Chain
Dx (g ◦ f ) = (Dx g ◦ f ) · Dx f
Dx (g(f (x)) = g 0 (f (x)) · f 0 (x)
Absolute Value
x
Dx |x| =
= sign x
|x|
Examples
1. What is the derivative of f (x) = sin(3x)?
For this problem, we use the chain rule. The inside function is 3x. The derivative of 3x is 3
(using the constant multiple and identity rules). The outside function is sin x. Its derivative is
cos x. We use the chain rule to compute the rest:
Dx (sin 3x) = cos(3x) · (Dx 3x)
= 3 cos 3x
2. Find the derivative f 0 of f (x) =
x2
For this, we use the quotient rule and power rule.
tan x
tan x(2x) − x2 sec2 x
tan2 x
2x tan x − x2 sec2 x
=
tan2 x
f 0 (x) =
3. Find the derivative of f (x) = sin2 x + cos2 x without using trigonometric identities.
Each term is a trigonometric function followed by raising to the 2nd power. Thus we use chain
rule for each term.
Dx (sin2 x + cos2 x) = 2 sin x(Dx sin x) + 2 cos x(Dx cos x)
= 2 sin x · cos x + 2 cos x(− sin x)
=0
Tangent Line of Function (with trig functions)
Recall that the tangent line to the graph of f (x) at an x-value c is the line with slope f 0 (c) passing
through the point (c, f (c)).
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Example
What is the tangent line to the function f (x) = sin x · tan x at the x-value π/6?
First, we find the derivative using the product formula:
f 0 (x) = sin x · Dx (tan x) + (Dx sin x) · tan x
= sin x · sec2 x + cos x · tan x
Now we find the slope of the tangent line:
f 0 (π/6) = sin π6 · sec2 π6 + cos π6 · tan π6
2 √
1
3 1
2
√
√
+
= ·
2
2
3
3
7
=
6
And to find the point (c, f (c)):
√
f (π/6) = sin π6 tan π6 =
3
6
Now we use the point slope formula
y − y1 = m(x − x1 )
to find the equation of the line:
√
y−
7
π
3
=
x−
6
6
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Implicit Differentiation: Tangent Line of Curve (with trig
functions)
This is the same story as in the case of a tangent line to a curve, except now you use implicit
differentiation to find the slope of the tangent line. Implicit differentiation will give you the slope
dy
dx in terms of x and y.
Example
Consider the curve C cut out by the equation
2 cos(4x) +
20 2 2
x y =5
π2
and the point P = (π/8, 4) on C.
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What is the tangent line to C at P ?
First we implicitly differentiate:
2(− sin 4x) · 4 +
20
(2xy 2 + 2x2 yy 0 ) = 0
π2
20
(2xy 2 + 2x2 yy 0 ) = 8 sin 4x
π2
40 2 0
40
x yy = 8 sin 4x − 2 xy 2
π2
π
40x2 yy 0 = 8π 2 sin 4x − 40xy 2
π 2 sin 4x y
−
5x2 y
x
dy
π 2 sin 4x y
=
−
dx
5x2 y
x
y0 =
Now we evaluate
dy
dx
at P
π 2 sin π2
dy 4
32 2π
−
=− +
≈ −8.92928
=
2
dx P
5(π/8) · 4 π/8
π
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Plugging in for point slope formula, we get the equation
32 2π π
y−4= − +
x−
≈ −8.92928(x − π/8)
π
5
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Related Rates (with trig functions)
Blueprint for how to solve a related rate problem:
(1) Draw a picture! (If you can, or if it hasn’t been made for you.)
(2) What are all relevant, non-constant variables? (i.e., those things which are changing with time)
(3) What rates are given?
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(4) What is the deisred rate - i.e. the rate we want to know? And under what condition?
(5) What are all possible relations between the variables from step 2?
(6) Implicitly differentiate each relation from step 3 to obtain relations between the rates.
(7) Box or circle the following things:
• The given rate from step 3.
• The condition(s) from step 4.
• Each relation between variables from step 5.
• Each relation between rates and variables from step 6.
(8) The problem is now just basic (albeit complicated) algebra: solve for the desired rate.
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