x = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)

Transcription

1 Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane limit at left equals M. Notation: lim M c We say that f has a right-hane limit at right equals N. Notation: lim N + c Two-Sie Limits Suppose a function f is efine near but not necessarily at If the left-hane limit c. c Stuy Sheet (7.) c of M if the limit of f () as approaches c from the c of N if the limit of f () as approaches c from the c. lim L an the right-hane limit lim R are both equal to the c same number M (an say that the two-sie limit at c eists an is equal to M ). + c Limits of rational functions A rational function is a quotient of polynomials. It will have the form where f an g are polynomials (g 0)., g( ) Consier f (). In this function we have two eamples of limits involving infinity First, we observe: lim 0 Here infinity is involve as we fin the limit of the function as approaches zero from the left. In reality, when the answer to a limit problem is infinity, we are really saying that there is no limit. The negative infinity answer oes tell us that the value of the function is an etremely large number. This is better than simply saying there is no limit. Note: when f ( ) or f ( ) then the graph of the function must have a lim a vertical asymptote. We also see that: lim + 0 lim a In this case, we see that the right-han limit as approaches zero is positive infinity. We now know that officially: lim f ( ) oes not eist because the left-han an right-han limits o not agree. 0 We can also observe that: Here is another way that infinity is ealt with in limits. We can take limits as taking a limit of the type: f ( ) an the answer is a constant. Then the function has a lim horizontal asymptote at the function value.. When

2 Stuy Sheet (7.) Definition of Continuity A function f is sai to be continuous at the point a if each of the following conitions is satisfie: () () Continuity an Discontinuity If f is continuous at every real number a, then f is sai to be continuous. If f is not continuous at a, then f is sai to be iscontinuous at a. The function f can be iscontinuous for two istinct reasons: () Removable iscontinuity () Nonremovable iscontinuity (i)jump Discontinuity Definition of Continuity on a Close Interval Let f be efine on a close interval [ a, b]. If f is continuous on the open interval ( b) a, an then f is continuous on the close interval [ a, b] Continuity of Polynomial an Rational Functions (a) Any is continuous everywhere; that is, if it is continuous on (b) Any is continuous everywhere it is efine; that is, it is continuous on Continuity of Some Other Functions The following type of functions are continuous at every number in their omains: () () (3) (4)

3 Stuy Sheet (7.3) Tangent Line to a Graph Secant Line to a Graph (, y) (, y) f If ) an ) be two orere pairs of a function, then the slope of the function f with respect to over an (, y (, y interval is approimate by Slope Change in Change in Δ Δ. Difference Quotient The epression f ( + Δ) Δ, Δ 0 is calle the ifference quotient. Definition of the Slope of a Graph The slope m of the graph of f at the point (, ) is equal to the slope of its tangent line at (, ) by, an is given m lim0 Δ f ( + Δ) Δ, provie this limit eists.

4 Definition of the Derivative For any function f, the erivative function is efine as follows: A function f is ifferentiable at if eists at. Differentiability Implies Continuity If f is ifferentiable at a, then Consier four cases in which a function f is NOT ifferentiable.

5 Stuy Sheet (7.4) Pascal s Triangle Pascal s Triangle provies us with a short-cut for fining the coefficients of the terms that result when we raise a binomial to some power. ( + Δ) ( ) + Δ ( ) 3 + Δ 3 3 ( ) 4 + Δ ( ) 5 + Δ ( + Δ) 6 Eample: ( ) 6 + Δ ( ) Δ + ( ) Δ + ( ) Δ + ( ) Δ + ( ) Δ + ( ) Δ + ( ) Δ Definition of the Derivative: f '( ) lim0 Δ f ( + Δ) Δ Problem: We will eplore the relationship between some familiar power functions an their erivatives.. Use the efinition of the erivative to fin the erivative of.. Repeat part for Repeat part for 4.

6 n 4. Let us try to generalize this process for a function, where n is a positive integer. n ( + Δ) f '( ) lim Δ 0 Δ n In fact, it is true for any positive integer n, ( n ) n n This is calle the Power Rule in ifferential calculus. Eamples: Differentiate each function using the power rule. a) 7 b) g ( ) 0 Now consier the cases in which n is a negative integer, for eample,,, 3 etc? Problem: Once again, we will use the efinition of the erivative to get the erivatives of each of these functions.. (a) Fin the erivative of. (b) Fin the erivative of. (a) Fin the erivative of (b) Fin the erivative of using the power rule.. using the power rule. So it appears that the power rule will also work when n is a negative integer. Eamples: Differentiate each function using the power rule. a) 3 b) g ( ) 4

7 3 Now consier the cases in which n is a fraction, or. If we were to use the power rule to get the erivative of we woul write: an then, applying the power rule, we woul get '. Problem: Fin the erivative of using the efinition of the erivative. In fact, (although we have not proven it to be true) the power rule works when n is any real number, an we shall assume this to be the case throughout the remainer of our work. Net, we will take the erivative of sums or ifferences of such terms: e.g. + an 3 5. We will also take the erivatives of power functions that are precee by constant multipliers; terms such as 4 an. The Constant Multiple Rule (The erivative of a constant times a function is the constant times the erivative of the function.) 3 Eample: [ cf ()]. The Sum Rule (The erivative of a sum is equal to the sum of the erivatives.) [ + g( ) ] Eample: ( + ) The Difference Rule (The erivative of a ifference is equal to the ifference of the erivatives.) 3 [ g( ) ] Eample: 5 4. Derivative of A Constant Function (The erivative of a constant function is always.) (c) Eample: (π ) 5. Derivative of A Linear Function (The erivative of a linear function is just its.) [ m + b] Eample: ( + 5)

8 Eponential Functions Consier the function f ( ) a,where a is a constant. First, let us fin the erivative of f ( ) using the efinition of a erivative. f '( ) Net, let us fin the erivative of f 3 ( ) using the efinition of a erivative. f '( ) Derivative of the Eponential Function: ( a ) a ln a Use a similar proceure to that in the previous to fin the erivative function for f ( ) e. Derivative of the Natural Eponential Function: ( e ) e

9 Stuy Sheet (7.5) Average Rate of Change ( b, f ( b)) ( a, f ( a)) f If ( a, f ( a)) an ( b, f ( b)) be two orere pairs of a function, then the slope of the function f with respect to over an interval is approimate by Average Rate of change Change in Change in Δ Δ. Position Function A position function s (t) is a function that gives the irecte or signe istance of an object from a fie reference point calle the origin, 0. In this contet, t has the units of time an s the units of istance. Average Velocity For the position function s (t), the average velocity over a time interval a t b is efine by Change in is tan ce Change in time Δs Δt s( b) s( a) b a Difference Quotient

10 The epression f ( + Δ), Δ 0 is calle the ifference quotient. Δ f Definition of Instantaneous Rate of Change The instantaneous rate of change of y f () as is the limit of the average rate of change on the interval [ + Δ],, as Δ approaches 0. Δ f ( + Δ) Δ lim 0, Instantaneous Velocity For the position function s (t), the instantaneous velocity v at a, or v (a), is given by s( a + Δt) s( a) lim Δt 0 Δt S a

11 Stuy Sheet (7.6) Prouct Rule: If f an g are both ifferentiable, then [ g( ) ] Quotient Rule: If f an g are both ifferentiable, then g( )

12 Stuy Sheet (7.7) Chain Rule: I f ( g( ) ) F, then F '( ) f '( g( )) g'( ) In Leibniz notational form: If y f (u) an u g(), then y f ( g( )) an y y u. u Let us eplore why the Chain Rule works: Problem: If Yung can run twice fast as Uri an Uri can run three times as fast as Xiomara, then how many times faster than Xiomara is Yung? Now let the istance that Yung runs in some fie time be Δ Y, the istance that Uri runs in that same amount of time be Δ U, an the istance that Xiomara runs in that same amount of time be Δ X. Then Δ Y ( ) Δ U Δ U ( ) Δ X. So Δ Y ( )( ) Δ X Another way to epress this is, THIS IS THE ESSENCE OF THE CHAIN RULE. Consier y u an + u. y is a function of u, an u is a function of. Let s f ( u) u an ( ) + u. Then y y u u