Create Presentation
Download Presentation

Download Presentation

Graph the 6 trig functions using your calculator and make a sketch from

Graph the 6 trig functions using your calculator and make a sketch from

107 Views

Download Presentation
Download Presentation
## Graph the 6 trig functions using your calculator and make a sketch from

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**ET 1.4a**Graph the 6 trig functions using your calculator and make a sketch from What trig identities do you need to know to enter sec(x) csc(x) cot(x) Reciprocal Identities**y= cos x**y= sin x y= sec x y= csc x y= cot x y= tan x**Find the limit (if it exists). If it does not, explain why.**REMINDER OF DEF’N OF LIMIT (1.3 Notes – Don’t recopy) Book: DNE because decreases without bound Book: DNE because increases without bound Let f be a function and let a be a real number. exists iff Book: DNE because left & right limits don’t exist 1. Other answers: Left DNE Right DNE L not equal to R 2. 3.**Edwards’ definition of continuity:**Don’t need to lift your pencil to draw the function. Causes of discontinuity: The road has been “hijacked” hole infinity jump (Removable Disc: canceling) Formal definition of continuity at x = a Note: Polynomials are always continuous.**a**a a f(a) defined Limit Exists But f(a) not defined Limit Exists f(a) defined Limit at a DNE A function is continuous at a if: Yes Notice: The def’n removes of all of the above trouble spots. Are there other intervals on the graphs above that are continuous?**Greatest Integer Function: Check for Understanding**Removable discontinuity? Continuous? NO Restated: Can you remove the discontinuity by simply redefining one point? NO Limit Exists Everywhere? NO**Check for Understanding.**NO Continuous Everywhere? YES Continuous Over Domain? Removable discontinuity? NO Restated: Can you remove the discontinuity by simply redefining one point? NO Limit Exists Everywhere?**Check for Understanding.**NO Continuous Everywhere? YES Continuous Over Domain? Removable Discontinuity? NO Restated: Can you remove the discontinuity by simply redefining one point? YES Plug the hole… Redefine the function using piece wise. Limit Exists Everywhere? How could you fix this discontinuity?**Check for Understanding.**NO Continuous Over Domain? Removable Continuity? YES Restated: Can you remove the discontinuity by simply redefining one point? Limit Exists Everywhere? YES**Discuss the continuity of the function.**x = - 8 x = -2 #1, #3 #3 x = 0 x = 3 #2, #3 #1, #2, #3 Where is the function not continuous? Using the def’n of limit, why?**Find the x-values (if any) at which f is not continuous.**Which of the discontinuities are removable? x = 0 x = - 1 x = 1 Discontinuous at: What happens at the other discontinuities? What makes it removable? Canceling Infinity: Asymptotes**Continuity on Open vs. Closed Interval.Trick is to look at**the end points. Continuous on (a, b) Not Continuous on [a, b] Continuous on (a, b) Continuous on [a, b] If we close these holes then we will have continuity on the closed interval [a, b]**Limit Properties vs.**Continuity Properties SIDE BY SIDE VERY SIMILAR “If you start with good stuff, you’ll end up with good stuff.” True about f(g(x)) too!**Continuous at every point in theirdomain.**are always continuous. • POLYNOMIAL • 2. RATIONAL • 3. RADICAL • 4. TRIGONOMETRIC Continuous provided denominator Continuous as long as radicand > 0 Continuous as long as not located at an asymptote.**Assignment 1.4a**3, 5, 7, 17, 21, 27, 29, 47, 49, 63, 69, 77, 79, 95, 103**ET 1.4b: Given s(t)= -16t2+500**If a construction worker drops a wrench from a height of 500 feet, how fast will the wrench be falling at exactly 2 seconds? Let t = 2 Then s(2) = -16(2)2 + 500 = 436 Select (a, f(a)) a t Calculate slope of secant to estimate slope of tangent. 500 s(t) s(a) Compare to how you solved 1.3: #103 pg 69 t a t 2 t 2 (2) 2 + 2 = -64 ft/sec t 2**GREATEST CONSEQUENCE OF CONTINUITY.**Intermediate Value Theorem Simply put… If a stalk of corn was 1 foot tall on a given day and then measured some time in the future at 5 feet, then the intermediate value theorem guarantees that the corn stalk must’ve been 4ft tall at some point and time. Insert picture of corn stalk. Reason: The growth of the stalk is continuous.**The Intermediate Value Theorem often can be used to locate**zeros of a function that is continuous on a closed interval. • Specifically, if f is continuous on [a, b] and f(a) and f(b) differ in sign, the Intermediate Value Theorem guarantees the existence of at least one zero of f in the closed interval [a, b] .**Given some function f that is continuous on the closed**interval [0, 1] If f(0) = negative f(1) = positive Then there exists x = c such that f(c) = 0 Intermediate Value Theorem, which relies on continuity, allows us to use the bisection method to find the zeros.**Assignment 1.4b**22, 30, 50, 72, 76, 82, 87, 91, 92, 108