1. Consider f(x) = x 2 What is the slope of the tangent at a=0?

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Presentation transcript:

1. Consider f(x) = x 2 What is the slope of the tangent at a=0?

Consider f(x) = x 2 So slope at a local max or min of a smooth curve is zero.

2. f is differentiable at a for which graph? a a a a a A. B. C. D. E.

2. f is differentiable at a for which graph? ANSWER: B a a a a a A. B. C. D. E.

3. Which graph(s) have a corner, but not a cusp at x = a? a a a a a A. B. C. D. E.

3. Which graph(s) have a corner, but not a cusp at x = a? ANSWER: A and E a a a a a A. B. C. D. E.

4. For graph A, what is the left- hand derivative at x = a? a a a a a A. B. C. D. E.

4. For graph A, what is the left- hand derivative at x = a? ANSWER: -1 a a a a a A. B. C. D. E.

5. For graph E, what are the left-hand and right-hand derivatives at x = a? a a a a a A. B. C. D. E.

5. For graph E, what are the left- hand and right- hand derivatives at x = a? ANSWER: 0 and -∞ a a a a a A. B. C. D. E. (At vertical tangents, as with limits, if we write f ’ (a) =  it means the slope “approaches  ”, but that is not a real number slope.)

6. For graph D, what is the right- hand derivative at x = a? a a a a a A. B. C. D. E.

6. For graph D, what is the right- hand derivative at x = a? ANSWER: dne there is no derivative at a discontinuity ! a a a a a A. B. C. D. E.

When is f(x) NOT differentiable? Where it is not continuous Where it is not continuous (but you can count a one-sided derivatives at closed endpts of closed intervals) Where f ’ is not defined Where f ’ is not defined At a corner At a corner At a cusp At a cusp At vertical tangents At vertical tangents (As with limits, if we write f ’ (a) =  it means the slope “approaches  ”, but that is not a real number slope.) Corners Cusp Vert. Tangents

7. Write f(x) = | x | as a piece-wise defined function.

7. Write f(x) = | x | as a piece-wise defined function. ANSWER:

8. For f(x) = | x |, find f ‘ (0).

8. For f(x) = | x |, find f ‘ (0). ANSWER: m = -1 m = 1

Consider f(x) = | x | Is f(x) differentiable at a = 0? NO! It is continuous, but not differentiable! This is a CORNER.

9. Must get all three correct to count the point. T F If a function f is differentiable at a, then it is continuous at a. T F If a function f is continuous at a, then it is differentiable at a. T F If a function f has a limit at a, then it is continuous at a.

9. Must get all three correct to count the point. T If a function f is differentiable at a, then it is continuous at a. F If a function f is continuous at a, then it is differentiable at a. F If a function f has a limit at a, then it is continuous at a.

Definition A function f is differentiable on an open interval if f ’ (x) exists for every x in that interval. (i.e. left & right slopes are equal) A function is differentiable on a closed interval [a, b] if it is differentiable on the open interval (a, b) and if the following (one-sided) limits exist: On the right side of the lower bound. On the left side of the upper bound. Can be called the right-hand derivative at a. Can be called the left-hand derivative at b.

At a closed endpoint of an interval: (not elsewhere) We accept the one-sided derivative as an overall derivative. I.E: No derivative at an open endpt, but may be differentiable at the closed endpt [a, b) [a,  ) (a, b] (- , b] If a is in an open interval (meaning in the middle of some interval), then f ’ (a) exists, IFF, both right-hand and left-hand derivatives at a exist and are equal.

10. Must get all three correct to count the point. T F If the left hand deriv at x = a is ∞, then (a, f(a) ) is a cusp. T F If the left hand deriv at x = a is 5 and the right hand deriv at x = a is - 5, then (a, f(a) ) is a cusp. T F A piece-wise defined function contains a corner.

10. Must get all three correct to count the point. F If the left hand deriv at x = a is ∞, then (a, f(a) ) is a cusp. F If the left hand deriv at x = a is 5 and the right hand deriv at x = a is - 5, then (a, f(a) ) is a cusp. F A piece-wise defined function contains a corner.

There is a CORNER at P(a, f(a)) if f is continuous at a and: if right & left-hand derivatives exist at a but are unequal or if ONE of those derivatives exists at a and f ’ (x)  ±  as x  a + or as x  a –

CORNER at P(a, f(a)) if f is continuous at a and: if right & left-hand derivatives exist at a but are unequal or if ONE of those derivatives exists at a and f ’ (x)  ±  as x  a + or as x  a – Derivatives exist, but are unequal. left-derivative exists, but right derivative goes to - .

11. Do this one in your notebook to keep for reference. Use the def of derivative to find f ‘ for

11. Use the def of derivative to find f ‘ for

12. Find f ‘ (9) and f ‘ (0) for

Definition The graph of a function has a vertical tangent line x = a at the point P(a, f(a)) if f is continuous at a (use one-sided limit for cont if it is an endpt) and if

Some more vertical tangents

Definition The graph of a function has a CUSP at the point P(a, f(a)) if f is continuous at a and if the following 2 conditions hold: i) f ’ (x)   as x approaches a from one side ii) f ’ (x)  -  as x approaches a from the other side Look for the “seagull”