MATH 127 MIDTERM 2
Mar 23, 2016
MATH 127 MIDTERM 2
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
Tutor: Maysum Panju
• 3B Computational Mathematics
• Lots of tutoring experience
• Interests: – Harry Potter– Pokémon– Calculus
3
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
4
Basic Derivative Rules
5
Memorize Them.
Basic Derivative Rules
• The following derivative rules should be memorized:
Sum Rule
Scalar Rule
Quotient Rule
Product Rule
Power Rule
6
The Chain Rule
• The following derivative rule should also be memorized:
Chain Rule
7
Exponential Derivatives
• Derivative of Exponentials: – Slope is proportional to height!
8
Logarithm Derivatives
• Derivative of Logarithms: – Slope is proportional to 1/height!
9
Trig Derivatives
• The derivative of a wave is another wave.• The derivative of anything else (trig) is
somewhat uglier.
10
Inverse Trig Derivatives
• It’s easiest to derive inverse functions using implicit differentiation.
11
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
12
Rate of Change Applications
13
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.
14
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?
15
Solution to Ladder Problem
Given:
Unknown:
We can identify the main relating equation:
16
Solution to Ladder Problem
Implicitly differentiate the main equation:
17
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
18
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 .
19
20
Break Time…
Derivatives and Graphs
21
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
22
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).
23
Concavity and Inflections• Concave up: the derivative is increasing.• Concave down: the derivative is decreasing.
• Point of Inflection: change in concavity.
24
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.
25
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
26
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!
28
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)
29
Optimization
• The objective to maximize is• The constraint is
i.e.
• So the objective is
• Differentiate:
• Set to 0, solve: and
30
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!
31
Curve Sketching
• Sketch the curve .
32
Other Uses of Derivatives
33
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:
34
Newton’s Method Example
35
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
36
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!
38
Questions and Practice Problems
39
Monster Example
• Compute the derivative of y:
Deceptively simple…40