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MATH 127 MIDTERM 2

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

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MATH 127 MIDTERM 2

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  1. MATH 127 MIDTERM 2

  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 Location Cusco, Peru # Students 22 Project Cost $16,000

  3. Tutor: Maysum Panju • Maysum doing Calculus during a Spelling Bee. • 3B Computational Mathematics • Lots of tutoring experience • Interests: • Harry Potter • Pokémon • Calculus

  4. 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

  5. Basic Derivative Rules Memorize Them.

  6. Basic Derivative Rules • Power Rule • Sum Rule • Product Rule • Scalar Rule • Quotient Rule The following derivative rules should be memorized:

  7. The Chain Rule • Chain Rule The following derivative rule should also be memorized:

  8. Exponential Derivatives • Derivative of Exponentials: • Slope is proportional to height!

  9. Logarithm Derivatives • Derivative of Logarithms: • Slope is proportional to 1/height!

  10. Trig Derivatives The derivative of a wave is another wave. The derivative of anything else (trig) is somewhat uglier.

  11. Inverse Trig Derivatives It’s easiest to derive inverse functions using implicit differentiation.

  12. Implicit Differentiation • Can’t solve for y:Don’t despair! • Differentiate wrtx:Use chain rule! • Product Rule • Chain Rule • Solve for dy/dx:Always easy! You can use the chain rule to differentiate even when you can’t solve for y explicitly!

  13. Rate of Change Applications

  14. 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. Figureout what relates to what, and how. • Differentiate implicitly. • Substitute variables until you can solve for unknown.

  15. 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?

  16. Solution to Ladder Problem Unknown: We can identify the main relating equation: Given:

  17. Solution to Ladder Problem Implicitly differentiate the main equation:

  18. 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

  19. 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 .

  20. Break Time…

  21. Derivatives and Graphs

  22. Derivatives and Graphs A derivative describes the rate of change of a graph. This tells us the shape of our graph.

  23. 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 changesfrom increasing to decreasing (or decreasing to increasing), we have a maximum(or minimum).

  24. Concavity and Inflections • Concave Up f ’’ > 0 • Concave Downf ’’ < 0 • Concave Downf ’’ < 0 • Concave Up f ’’ > 0 • Inflection:f’’ = 0 Concave up: the derivative is increasing. Concave down: the derivative is decreasing. Point of Inflection: change in concavity.

  25. 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.

  26. Maximum/Minimum Values • f’(x) exists • f’(x) does not exist • f’(x) = 0 • Domain of f: • x a min • x a min • x a max • x a max • Critical Points • 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.

  27. How to tell Max or Min? • Concave Down: Max • Concave Up: 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 amaximum or minimum!

  28. 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!

  29. 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)

  30. Optimization The objective to maximize is The constraint isi.e. So the objective is Differentiate: Set to 0, solve: and

  31. Curve Sketching Steps Find the domain of f. Find the x and yintercepts. 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!

  32. Curve Sketching • Sketch the curve .

  33. Other Uses of Derivatives

  34. 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:

  35. Newton’s Method Example

  36. Newton’s Method Example • This is equivalent to finding the positive root of which has • Start with a guess of 9. • We get compare with Estimate using one round of Newton.

  37. L’Hôpital’sRule If or (and f’, g’ both exist), then Sometimes, manipulate the expression to get it in this form. Example: Show that .

  38. 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!

  39. Questions and Practice Problems

  40. Monster Example Deceptively simple… Compute the derivative of y:

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