MATH 140 Lecture Notes - Lecture 12: Differentiable Function, Farad

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greywater-buffalo920

23 Jul 2013

School

McGill University

Department

Mathematics & Statistics (Sci)

Course

MATH 140

Professor

Ewa Duma

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MATH 140 Full Course Notes

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We previously considered the derivative of a function f at a xed number a : We can change the value of a and replace it with x, we will obtain: Figure 2 f"(a) = lim (h > 0) [f(x+h)-f(x)]/h (b) Chapter 2 limits and derivatives f x x 3 x. , nd a formula for (b) illustrate by comparing the graphs of. Which is useful to us as we attempt to nd the derivative of higher functions. Example 2 function f" is called the derivative of f because it has been derived from f by the limiting operation. has horizontal tangents and. Example: if f(x) = x^3 - x, nd a formula for f"(x) using the de nition of a derivative. (b) we use a graphing device to graph f these graphs serve as a check on our work in part (a).

Related Questions

1. Use the definition of continuity and the properties of limitsto show that the function is continuous at the given number(a).

f(x) = (x+3x^5)^4 , a= (-1)

lim f(x) as x approaches -1 (x+3x^5)^4

= (lim x +3 times lim x^5)^4

= (-1 +3 (???)^5)^4 ------got stuck here

= ???

f(-1) = ??

2. Find the average value f(ave) of the function f on the giveninterval

f(x)= 6x -x^2 , [0, 4]

3. Find the area of the shaded region between a curve and aline:

y = 9x-x^2 and y = 2x

the functions intersect at (0,0) and (7, 14)

I need to figure out the area of the upper half, above the y =2xline

it is a concave down parabola

4. (16 pts) MULTIPLE CHOICE! CIRCLE THE CORRECT ANSWER IN EACH PART. (I) ( 4 pts) Complete the statement of the Intermediate Value Theorem: Suppose f is continuous on the interval [a, b] and L is a number between f (a) and f (b). Then there is at least one number c in (a, b) such that (a) C = L; (b) f(c) = 0; (c) f(c) = L; (a) f' (c) = L; (e) C = 0; (f) f (a) < c < f (b); (g) NONE OF THE PREVIOUS ANSWERS. (II) (4 pts) The figure shows the graph of a function f. At what point (points) c does the conclusion of the Intermediate Value Theorem hold for f (x) on the interval [1, 5] and L = 2. --- - - - -1 - - - 2 - - I- - - (a) c = 1; (b) C = 0; (c) C = 2; (a) c = 3; (e) C = 4; (f) C = 5; (g) c= -1. (III) (4 pts) Given that cos x s f (x) s et, for all x > 0 evaluate lim f (x) x+0+ (a) lim f (x) = 0; (b) lim f (x) = 1; (c) lim f (x) = e; x0+ X0+ (d) lim f(x) DOES NOT EXIST. WE DON'T HAVE ENOUGH INFORMATION TO ANSWER THE QUESTION *40+ (IV) (4 pts) Given the function 1 f (x) = , find its inverse, f+(x) x -2 (a) f (x) = x -2; (b) f«+ (x) = x+2 ; (c) fº+ (x) = -2; (f) f-+ (x) -2; (e) f (x) DOES NOT EXIST.

MATH 140 Lecture Notes - Lecture 12: Differentiable Function, Farad

MATH 140 Full Course Notes

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