math 127 midterm 2
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MATH 127 MIDTERM 2. 2010 Outreach Trip. Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects ELT Classroom enables better job prospects. More info @ studentsofferingsupport.ca/blog. Summary Date Aug 20 – Sept 4 - PowerPoint PPT PresentationTRANSCRIPT
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MATH 127 MIDTERM 2
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2010 Outreach TripSummaryDate Aug 20 – Sept 4Location Cusco, Peru# Students 22Project Cost $16,000
Building ProjectsKindergarten Classroom provides free
educationSewing Workshopenables better job prospectsELT Classroom enables better job prospectsMore info @
studentsofferingsupport.ca/blog 2
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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.
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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.
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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…
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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
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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
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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
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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