Pure Mathematics F2: Differentiation

30 flashcards · Mathematics · by lakesare

1. [Wording] What 2 words do we use to denote:

   we differentiate f The word 'differentiate' arises because the definition involves the differences f(x) − f(c) and x − c.the derivative of f The use of the root "deriv" in the context of differentiation seems odd: we have differentiation, differentials, differentiable, differential equations, and then for some odd reason, "derivative." Why/how did this happen?In French there is less of a problem with terminology: differentiation is called "dérivation", and derivative is, you guessed it, "la dér…

2. [Derivatives] Visually imagine the difference quotient of f at c.

   This expression is called the difference quotient for f at c. Rise over run!

3. [Derivatives] Visually imagine the gradient of f at c.

   This expression is called the gradient (or slope) of f at c. Limit of "rise over run"!

4. [Derivatives] What does Df(x) mean?

   (In Lagrange notation)

5. [Limits] Name 2 only cases when a limit does not exist?

   it tends to 2 different numbers it tends to ∞

6. [Derivatives] Name 2 ways to present difference quotients. Difference quotient is simply rise over run.

   To prove that a function is differentiable, it is usually most convenient to use this form of the difference quotient.

7. [Derivatives] Fundamentally - why is y not differentiable at 0?

   Because it oscillates infinitely often in any interval containing x=0.

8. [Derivatives] What does this stand for?

   Left derivative

9. [Derivatives] What does this stand for?

   Right derivative

10. [Derivatives] Given this definition of f'(c), what's the left derivative f'L(c)?

11. [Derivatives] Give an example of a function where the concepts of left derivative and right derivative are useful.

    Although we know that the function f (x) = |x| is not differentiable at 0, the graph of f shown in Figure 5 suggests that chords which join the origin (0, 0) to points (h, f (h)) have gradients equal to 1 if h > 0, and equal to −1 if h < 0. This example suggests the concept of a one-sided derivative.

12. [Derivatives] What the Glue Rule for Differentiation is about?

    About how to differentiate piecewise functions.

13. [Derivatives] What does it mean when we say that differentia bility is a local property?

    Depends on the behaviour of the function in any open interval (no matter how small!).

14. [Derivatives] Name an example of a continuous nowhere-differentiable function.

    The blancmange function.

15. What's the name of this dish?

    The word blancmange derives from Old French blanc (white) mangier (dish). Blancmange from French: blanc-manger [blɑ̃mɑ̃ʒe]) is a sweet dessert popular throughout Europe commonly made with milk or cream and sugar thickened with rice flour, gelatin, corn starch, or Irish moss (a source of carrageenan), and often flavoured with almonds.Some similar desserts are French chef Marie-Antoine Carême's Bavarian cream, Italian panna cotta, the Middle Eastern muhallebi, Chinese annin tofu.

16. Why is this called the blancmange function?

    Because it looks like the blancmange (white-dish) dessert. Blancmange is a sweet dessert popular throughout Europe.

17. [Derivatives] Name 2 ways how we can approach showing that a hybrid function f(x) is differentiable at 0?

    1. via the original definition (difference quotient Q(h)) Q stands for difference quotient2. via the Glue Rule The Glue Rule enables us to determine whether or not certain hybrid functions are differentiable at particular points, without using the definition of differentiability.

18. [Derivatives] Visually imagine Glue Rule for differentiation VS Squeeze Rule for limits.

    Glue Rule for differentiation Squeeze Rule for limits

19. [Derivatives] What kind of shit does the Glue Rule let us do?

    We can glue y = x and y = sin x, and it will be differentiable everywhere!

20. What's the interesting difference between the definitions of concepts: limit continuity differentiability

    limit - ε-δ definition continuity - ε-δ definition differentiability - just a particular limit! #TODO add pics

21. Name 2 ways to name this point.

    c is a stationary point f' vanishes at c

22. Visually imagine the Rolle's Theorem.

    Rolle's Theorem is an existence theorem; that is, it tells us that a point c with the stated property exists, but not how to find it. Often it is difficult to evaluate c explicitly, and there may be more than one point c in (a, b) at which f' vanishes.

23. Visually imagine Rolle's Theorem VS Mean Value Theorem.

    Rolle's Theorem The Mean Value Theorem When f (a) = f (b), the Mean Value Theorem reduces to Rolle's Theorem.

24. Why is the Mean Value Theorem called so?

    The name comes from the fact that, due to the fundamental theorem of calculus, an average rate of change over an interval may be viewed as an average (or mean) of the instantaneous rates of change along the interval.

25. How would you prove the Mean Value Theorem using the Rolle's Theorem?

    Find the equation of the vertical distance h(x). By Rolle's Th, h'(x) = 0 somewhere. Solve h'(x)!

26. What's the interior of the interval I?

    The set of points in I which are not endpoints of I. We can write it as (I) [my own notation].

27. Visually imagine how would we prove the Increasing-Decreasing Theorem: if f'(x) ≥ 0, then f is increasing. The Increasing-Decreasing Theorem is also known as the 1st derivative test.

    Let x1, x2 be in I and x1 < x2 Apply the Mean Value Theorem on [x1, x2] We know f'(x) is positive → we know the chord is positive.

28. What are the 1st derivative test VS 2nd derivative test about?

    1st derivative test - increasing/decreasing The first-derivative test examines where the function is increasing or decreasing.2nd derivative test - local max/min

29. Name 2 steps to prove the inequality using differentials.

    1. Let f(x) = g(x) – h(x) We want f(x) > 0.2. Find all stationary points, and make sure they are > 0.

30. Visually imagine the L'Hopital's Rule.