Topic 5 Graph Sketching Reasoning Limits Dr J

Topic 5: Graph Sketching, Reasoning & Limits Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 31 st August 2015

Slide Guidance ? Any box with a ? can be clicked to reveal the answer (this works particularly well with interactive whiteboards!). Make sure you’re viewing the slides in slideshow mode. For multiple choice questions (e. g. SMC), click your choice to reveal the answer (try below!) Question: The capital of Spain is: A: London B: Paris C: Madrid

ζ Topic 5 – Graph Sketching and Reasoning Part 1: Limits

RECAP: Differentiation by 1 st Principles ? ? The “lim” bit means “what this expression approaches as h tends towards 0” δy δx ? ? ? The h disappears as h tends towards 0.

Algebra of Limits ? ? ? ?

Examples Useful in MAT! Bro Hint: Can we use that last law of limits? Common Sense Reasoning Way ? Formal Way ?

Examples Use your knowledge of the relative growth rate of various functions to find these limits, or state that they are divergent. ? ?

Indeterminate Forms ?

How could this go wrong? How should we do it? But there’s other times where we couldn’t do this:

l’Hôpital’s Rule This just means that if when we differentiate the numerator and denominator, we no longer have an indeterminate form. Example ? *Remember that ‘quotient’ is just a posh word for the result of a division. Hence why we have the ‘quotient rule’.

Test Your Understanding ? ? (Wolfram. Alpha. com can show you the step-by-step proof for these! e. g. http: //www. wolframalpha. com/input/? i=lim%5 Bx-%3 E 0%5 D%28 x+ln%28 x%29%29 You have to usually pay a subscription for the ‘proof’ feature, but if you buy the app for tablet/phone for a less than 2 quid, it has this feature without subscription!)

? ? In that second question, we cleverly turned a product (which was indeterminate) into a quotient (which was indeterminate), in order to apply l’Hopital’s ? Rule. Can we apply similar tricks to other kinds of expressions, to get a quotient? ? ? ? …As we used above

Example ? We can move log inside the limit. ? Use law of logs. ? We proved this earlier. ? ?

Bonus Question ? Laws of logs. ? ? Apply l’Hopital’s and simplify. Wow. It’s Euler’s constant! ?

ζ Topic 5 – Graph Sketching and Reasoning Part 2: Sketching Fundamentals

Graph Features? ? y-intercept? ? Turning ? Points? Roots? ? Asymptotes? ? An asymptote is a straight line that a curve approaches at infinity.

Graph Features? You may not have covered the following terminology yet: -1 ? ?

The two main ways of sketching graphs 1 Thing about the various features previous discussed. 2 And/or consider the individual components of the function separately, and think how they combine. We’re multiplying these two individual functions together. What happens at the peaks and troughs of the sin graph? Try sketching it!

Values to a power ?

Values to a power

3 D graphs ? Explanation: ?

Composite functions ?

Composite functions ?

Reciprocal functions

Functions transforms Click to view transformation 1 ? Click to view transformation -2 ?

Putting it all together… A [Source: Oxford MAT 2007] B C D

A harder one For this graph, it might be helpful to think about: 1. How do deal with the y 2. 2. The asymptotes (both horizontal and vertical). 3. The domain of x (determine this once you’ve dealt with the y 2). 4. Roots.

Not defined for -1<x<1. As x becomes larger, the +1 and -1 has increasingly little effect, so y = 1 for large x. Repeated above and below x-axis because we have y = √. . .

ζ Topic 5 – Graph Sketching and Reasoning Part 3: Reasoning about Solutions

Polynomials A polynomial of degree 4. Order 2 Quadratic Order 3 Cubic Order 4 Quartic Order 5 Quintic ? ? ?

Polynomials Can we generalise this to polynomials of any degree?

Polynomials of odd degree will always have a range which spans the whole of the real numbers. Polynomials of even degree will always have a finite range with a minimum or maximum. It goes ‘uphill’ if the coefficient of the highest-power term is positive. It will be ‘valley’ shaped if the coefficient of the highest-power term is positive.

Number of Roots Odd Degree Even Degree Minimum Roots: ? ? Maximum Roots: ? ?

Number of Roots Click to Start Animation Number of Distinct Roots: 01 2 3 4

Number of Roots a) b) ? ? ? ?

Repeated factors/roots ? ? ?

Repeated factors/roots Inflection point Touches axis Crosses axis ?

Turning Points max min ?

Points of Inflection However, when we have a point of inflection, then two of the turning points effectively ‘conflate’ into one. It’s a bit like having a max point immediately followed by a min (or vice versa)

Points of Inflection Stationary point of inflection Known as a ‘saddle-point’ Non-stationary point of inflection Known as a ‘non-stationary point of inflection’

Strategies for determining number of solutions METHOD 1: Reason about the graph As we already have done: • If it’s a polynomial, is the degree even or odd? • If we have a constant that can be changed, consider the graph shifting up and down. • We may have to find the turning points (by differentiation or completing the square) METHOD 2: Factorise (when possible!) ? METHOD 3: Consider the discriminant ?

Example [Source: Oxford MAT 2009] A B C D

Example

Example [Source: Oxford MAT 2009] A B C D

Example