math 127 midterm 2

MATH 127 MIDTERM 2

2010 Outreach TripSummaryDate Aug 20 – Sept 4Location Cusco, Peru# Students 22Project Cost $16,000

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Tutor: Maysum Panju

• 3B Computational Mathematics

• Lots of tutoring experience

• Interests: – Harry Potter– Pokémon– Calculus

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Maysum doing Calculus during a Spelling Bee.

Outline

• Derivative Rules• Rate of Change Applications

– Related rates, linear approximations • Derivatives and Graphs

– Shape of graphs, optimization, curve sketching• Other Uses of Derivatives

– Newton’s Method, L’Hôpital’s Rule, MVT

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Basic Derivative Rules

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Memorize Them.

Basic Derivative Rules

• The following derivative rules should be memorized:

Sum Rule

Scalar Rule

Quotient Rule

Product Rule

Power Rule

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The Chain Rule

• The following derivative rule should also be memorized:

Chain Rule

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Exponential Derivatives

• Derivative of Exponentials: – Slope is proportional to height!

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Logarithm Derivatives

• Derivative of Logarithms: – Slope is proportional to 1/height!

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Trig Derivatives

• The derivative of a wave is another wave.• The derivative of anything else (trig) is

somewhat uglier.

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Inverse Trig Derivatives

• It’s easiest to derive inverse functions using implicit differentiation.

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Implicit Differentiation

• You can use the chain rule to differentiate even when you can’t solve for y explicitly!

Can’t solve for y:Don’t despair!

Differentiate wrt x:Use chain rule!

Solve for dy/dx:Always easy!

Product Rule Chain Rule

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Rate of Change Applications

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Related Rates• Basic idea:

– A system is changing as time passes.– Different quantities change at different (but related) rates.– How fast does “X” change when “Y” (and “Z” and …) is

changing at rate “dY/dt” (and “dZ/dt” and …)?

• Steps…– Read problem. Draw diagram. Figure out what relates to

what, and how.– Differentiate implicitly.– Substitute variables until you can solve for unknown.

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Example: Falling Ladder

A ladder (5m long) leans against a wall. The bottom end moves away from the wall at a constant rate of 30 cm/s.

At what rate does the top of the ladder move down the wall when the bottom of the ladder is 4m away from the base of the wall?

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Solution to Ladder Problem

Given:

Unknown:

We can identify the main relating equation:

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Solution to Ladder Problem

Implicitly differentiate the main equation:

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Equation of Tangent Line

• What is the equation of the tangent line to the curve y=f(x) at x = a?

• Point slope form of a line:– If a line has slope m and passes through (x1, y1),

then the line has equation

• The tangent line has slope f’(a) and passes through (a, f(a))… So it has equation

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Linear Approximations• Strategy: If f is hard to compute at some point

x, then …– Find a nearby point (a) that is EASY to compute– Find the tangent line at a– Find the height of the line at x

• Example:– Approximate .

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Break Time…

Derivatives and Graphs

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Derivatives and GraphsA derivative describes the rate of change of a

graph. This tells us the shape of our graph.If the derivative is… Then the original graph is…

Positive IncreasingNegative DecreasingZero FlatIncreasing Concave upDecreasing Concave downFlat LinearLarge SteepSmall Nearly flat

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Increasing and Decreasing Intervals

• When a differentiable curve is increasing, the derivative is positive.

• When a differentiable curve is decreasing, the derivative is negative.

• When a differentiable curve changes from increasing to decreasing (or decreasing to increasing), we have a maximum (or minimum).

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Concavity and Inflections• Concave up: the derivative is increasing.• Concave down: the derivative is decreasing.

• Point of Inflection: change in concavity.

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Concave Down f ’’ < 0

Concave Up f ’’ > 0

Concave Down f ’’ < 0

Concave Up f ’’ > 0 Inflection:

f ’’ = 0

Maximum/Minimum Values

• At any point in the domain, either the curve is differentiable or it isn’t.

• If a differentiable point is a max/min value, the curve MUST be flat!

• If a curve isn’t differentiable at a point, then it may be a max or min... Can’t say anything.

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Maximum/Minimum Values

• So, to find max/min values…– Find all places where f is differentiable (f’ exists)

• Of those, find where f’ = 0– Of those, check which are max and which are min.

– In the rest of the domain, f is not differentiable (in particular, endpoints of a closed interval)

• Check ALL of these points for possible max/mins.

f’(x) exists f’(x) does not exist

f’(x) = 0

x a minx a max x a minx a maxDomain of f:

Critical Points

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How to tell Max or Min?

• If f’(a) = 0, try the following: • First Derivative Test:

– If the derivative changes sign (“+ to –” or “– to +”) at a, then you have a maximum or minimum!

– Otherwise, neither max nor min.• Second Derivative Test:

– If f’’(a) < 0 or f’’(a) > 0, then you have a maximum or minimum!

Concave Down: Max Concave Up: Min 27

Optimization

• An application of finding max/min values.• Steps:

– Understand the problem. Draw a diagram.– Find the objective function f to optimize. Use a

constraint so that f depends on only one variable.– Solve the equation f ’ = 0.– Determine if you found a max or min.– Check other critical points!

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Optimization

• A manufacturer wants to produce cylindrical cans with a volume of 250 mL. What dimensions will minimize the amount of material required for a can? (1 mL = 1 cm3)

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Optimization

• The objective to maximize is• The constraint is

i.e.

• So the objective is

• Differentiate:

• Set to 0, solve: and

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Curve Sketching Steps

• Find the domain of f.• Find the x and y intercepts.• Check for symmetry. (Even/Odd/Periodic)• Check for asymptotes.

(Vertical/Horizontal/Oblique)• Find intervals where f is increasing/decreasing.• Find maxima/minima (check critical points).• Check concavity and inflection points.• Sketch the curve!

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Curve Sketching

• Sketch the curve .

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Other Uses of Derivatives

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Newton’s Method

• An iterative method for finding roots of a function.– Guess a root.– Find the tangent line there.– Find the x-intercept of the tangent line.

• This is your new guess!– Repeat.

• Formulaically:

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Newton’s Method Example

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Newton’s Method Example

• Estimate using one round of Newton. • This is equivalent to finding the positive root

of which has• Start with a guess of 9.

• We get compare with

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L’Hôpital’s Rule

• If or

(and f’, g’ both exist), then

Sometimes, manipulate the expression to get it in this form.

Example: Show that . 37

Mean Value Theorem• If f is continuous on [a,b] and differentiable on (a,b),

then for some c in (a,b), we must have

• So, if your average travelling speed is 20km/h, then at some instant, you must have been travelling exactly AT 20km/h!

• Maybe more than once!

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Questions and Practice Problems

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Monster Example

• Compute the derivative of y:

Deceptively simple…40