CH 2: Limits and Derivatives

Transcription

1 2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent line and the slope of the curve are the same at that point P 2 the slope of the tangent line can be approximated by choosing two points on the curve Closer the two points are, better the approximation is This approximation gives the average velocity between the two points 3 the slope of the curve at the point P gives the rate of change of the curve at that point 4 if the curve describes the distance of a moving object, then the slope at any particular point will tell the instantaneous velocity of the object at that point See Figure 5 page The limit of a function the limit is a number that a sequence of numbers get close to (potentially achieving that value) 2 For a function f(x), the limit at the point x = a is lim f(x) = L if the function f(x) approaches (or even attains) the value L as x approaches a 3 when finding a limit of an expression, you should first plug in the value to see if it exists: 3 x lim x 2 x + 2 = 4 4 not all limits are defined, for example lim does not exist as the function does not approach x a particular y-value In this case, we can talk about left limit: lim f(x) and a right limit lim f(x) For example lim = and lim + x x x + x = x 5 if the left and the right limit at a point are equal, then the function has a limit at that point The limit could be different than the value of the function at the point (see Example 7 page 93) 6 if the limit lim f(x) is or then the function has a vertical asymptote at x = a sin x 7 important limit: lim = (it will be proved in chapter 3) x x 8 As you plug in the value, you can get a nonzero number divided by, which will give you the limit to be or, and you need to use the left-hand and right-hand limit Ex : lim = since lim x2 x x x + Ex 2: lim DNE since lim x3 Ex 3: lim x x 2 = since lim x + x 2 x x 2 = = lim x 2 x = and lim x3 x + x 3 = = = lim x x 2

2 9 Example: f(x) = x+ Then lim f(x) = f() =, however x 2 lim f(x) = x x 2 will give an incorrect value here!!!) the next section will teach you how to evaluate limits 23 Calculating limits using the limit laws (a calculator If the limits lim f(x) and lim g(x) exits, then (a) lim[f(x) ± g(x)] = lim f(x) ± lim g(x) (b) lim[cf(x)] = c lim f(x) (c) lim[f(x) g(x)] = lim f(x) lim g(x) (d) lim f(x) g(x) = lim f(x) lim g(x), where lim g(x) 2 lim (f n (x)) = lim [(f(x))] n = [lim (f(x))] n 3 lim constant = constant, and also lim x = a 4 lim x n = a n, if n > 5 lim x p q = a p q as long as the p th -root is defined 6 lim n f(x) = n lim f(x), for all functions f(x) for which the n th -root is defined 7 if two functions are the same except at the point a where the limit is evaluated (ie f(x) = g(x) when x a), then lim f(x) = lim g(x) 8 if f(x) g(x), then lim f(x) lim g(x) 9 if f(x) g(x), then lim f(x) lim g(x) if f(x) g(x) h(x), then lim f(x) lim g(x) lim h(x) Particularly, if it happens that lim f(x) = lim h(x) = L for some constant L, then we also obtain lim g(x) = L This is called Squeeze Lemma or Squeeze Theorem 25 Continuity a function f is continuous at a number a if lim f(x) = f(a) That is to say that the left limit must equal the right limit and it must equal the value of the function at that point 2 otherwise, a function could be continuous only on one side: (a) a function f is continuous at a number a from the left if lim f(x) = f(a) (b) a function f is continuous at a number a from the right if lim f(x) = f(a) + 3 a function is continuous on an interval or set if it is continuous on every point in that interval/set 2

3 4 if two functions f and g are continuous, then for some constant c, the following combinations of functions are continuous as well: (a) f ± g (b) f g (c) f g, if g (d) constant multiple of either functions: cf and cg (e) composition of functions: f g and g f 5 there are some classes of functions that are continuous everywhere on their domains: (a) polynomials (b) rational functions (c) root functions (d) exponentials (e) logarithmic functions (f) x (g) sin x and cos x (but not tan x) (h) ANY COMBINATION OF THE ONES ABOVE IS CONTINUOUS ON ITS DOMAIN That is, you only need to check continuity at the points where the function has a zero denominator or more than one piece of a function combined (the piecewise defined functions like #6 2 in Section 25) 6 the removable discontinuity is the one where lim f(x) = lim f(x), but different from the + value of f(a) 7 the jump discontinuity is the one where lim f(x) lim f(x) + 8 the infinite discontinuity is the one where the function goes to infinity as x approaches a fixed value: lim f(x) = ± and lim + f(x) = ± 9 The intermediate value Theorem: If f is continuous on (a, b), then for each N [f(a), f(b)], there is c (a, b) such that f(c) = N This means that a function will take all the image values between the numbers f(a) and f(b) Particularly, if say f(a) < and f(b) >, the function will have a root in the interval (a, b) 26 Limits at Infinity: Horizontal Asymptotes If f is defined on some interval (a, ), then lim very large values x f(x) = L if f(x) approaches L as x takes 2 If f is defined on some interval (, a), then lim f(x) = L if f(x) approaches L as x takes x very small values (or large negatives) 3 examples: lim x x n = and lim x x = You can see this from the graph of n x n 3

4 4 limits at infinity: divide by the highest power of x in the denominator and use that lim x x n = if n (same for ) The numerator can be any nonzero number Similar ideas for 5 the following are not defined: ± 6 however, these are: + = = e = = = 27 Derivatives and Rates of Change the slope of the tangent line to a curve at some point a gives the derivative of the function at the point a (for distance, the derivative gives the instantaneous rate of change, ie the speed) More exactly: if f(x) is the curve, then the slope of the tangent line (if this limit exists) at the point (a, f(a)) is f(x) f(a) m = lim x a 2 another way of writing that is in terms of a little change h in the x-value: f(a + h) f(a) m = lim h h This formula can be obtained if h = x a in the one above it 3 And so, we define the derivative of f(x) at the point a to be exactly the above limit, if the limit exists: f f(a + h) f(a) (a) = lim, h h and it gives the instantaneous rate of change with respect to x of the function f(x) at the point a 28 The derivative as a function we now define the derivative at every single point versus just one single point We call this one the derivative function: f f(x + h) f(x) (x) = lim h h 2 for different values of x, this function gives us the derivative at that value 3 the derivative function is not always defined everywhere For example if f(x) = x, then the derivative function does not exist at x = 4 for the derivative to exist at some point, the function must be continuous and smooth, ie it cannot have corners like the absolute value, it cannot be discountinous neither can it have a vertical tangent line (in all of these cases there is no slope of the tangent line at that point) 5 a function f(x) is differentiable if its derivative f (x) exists 6 and so, f must be continuous in order for f to exist, but it is not sufficient That is: f differentiable f continuous 4

5 f continuous f differentiable (the counterexample is f(x) = x, which is one of the common counterexamples to differentiability) 7 the second derivative f (x) is the derivative of f (x) Similarly the 3rd derivative can be defined, and even higher orders For distance, the second derivative is the acceleration, which shows how the speed changes instantaneously 5