Pentel OLS17 JAPAN GA MATH 131 : CALCULUS I SPRING 2020 Lab 1: Tangents and Limits Instructions: Work together with your group to understand the following ideas and solve the following problems. If you do not finish the lab today, it must be completed and turned in at the start of class next Friday (January 24). The entire group, not just the scribe, is responsible to see that this happens. Problem 1. You've seen the word "tangent" used two different ways in math: to describe a particular trigonometric function, tan(x) = sin(x) cos(x) , as well as to describe a line that just barely touches a curve. Let's find out how these two ideas are connected. (a) Above is the unit circle with an angle 0 in standard position shown. Assuming that the point A is at the origin, (0, 0), what are the coordinates of B (hint: one is sin(0) and the other is cos(0))? B (b) How long are the line segments AB and AE? How long are AC and BC? (c) Why are triangles AABC and AADE similar? C E d) Convince the rest of your group that BC DE AC AE (where BC is the length of BC), and then use the val- ues you found for the lengths of these segments in part (b) to show that the length of DE is tan(0). (e) Thus DE has length tan(0). DE is also tangent to the circle at what point? The fact that this line segment is tangent to the circle and is why its length was called the tangent of 0, or tan(0). There are many ways to try and evaluate lim f(x): numerically (by plugging in values of x that are close to a and using these to approximate the value of the limit), graphically (by using a graph to observe the behavior of f(x) and estimate the value of the limit), and algebraically (by simplifying f (x) to the point where a can be plugged in for x). The rest of the lab deals with each of these methods. Problem 2. Fill in the following table for f(x) = sin (z) (be sure your calculator is in Radian mode). -.1 -.01 -.001 001 .01 . 1 f () Use the values in the above table to estimate lim - sin (x) Problem 3. Use the numerical method (like you did on the previous problem) to determine the following limits. Write down some of the values you use for r to test the limit as well as the value of the function at those points to justify your answers.Pentel DES17 JAPAN AUTOMATIC P (a) lim x2+6x+8 -2+ 2 (b) lim e-1/2 2 (c) lim (1 + * ) 1/ 2 write DNE. Problem 4. Use the below graph of y = f(x) to evalute the following limits. If the limit does not exist, lim f(x) = lim f(x) = tim f(2 ) = f (1) = lim f(x) = 2 -+ 2 - lim f(x) = lim f (x) = f ( 2 ) = lim f(x) = I +3 - lim f (x) = lim f (x) = f (3 ) = lim f (x) = lim f(x) = 2 + 4 5 lim f(2) = f ( 4 ) = Problem 5. Evaluate the following limits algebraically. lim 2 2 - 1 2 2 - x-2- x - 2 lim x - 2+ x - 2 2 2 - 1 lim x-2x- 2 lim 2-+0 205 + 5x2 lim (2 - 3) (2 - 1) 2 - 3) (2 + 1) x-1 (x - 1)(x+ 2) lim C +3- 2 ( 2 - 3) 2 lim (2 - 3) (2+ 1) lim (2 - 3)(2+1) I -+3+ x(x - 3) 2 1-+3 ac ( 2 - 3) 2 (1+ h)3 - 1 lim lim h -+ h t + - 2 2 + t 3t lim lim t-+0 Vt2 + 4 - 2 I-+0- lim lim C -+0 It - lim f(x) where f(x) = x2+1 ifx