MATH 260

Math 260 - Week 1 Lab Name: Part I – Limits The limit of a function is a way to see the value that the function approaches as a variable in that function gets close to but not necessarily equal to some other valueThe four ways we have looked at to find the limit of a function are:  direct substitution of the limiting number into the function  simplifying the function first then substitute (factor or rationalize) examine values of the function from the left and right of the limiting number  examine the graph of the function at the limiting value Category 1: Directions:Look at the examples below then answer question 1& 2. 1.) Which of the Rules or Processes , , ,  mentioned aboveis/are being used to find the limit in the Category 1 examples above ? 2.) For the following limit why can you not use the same process as the examples above ? Category 2: Directions:Look at the examples below then answer question 3& 4. 3.) Which of the Rules or Processes , , ,  mentioned aboveis/are being used to find the limit in the Category 2examples above ? 4.) If you graph a Category 2 problem, what feature will appear at the limiting value ? Category 3:Directions:Look at the examples below then answer question 5. 5.) Which of the Rules or Processes , , ,  mentioned aboveis/are being used to find the limit in the Category 3 examples above ? Category 4: Directions:Look at the examples below then answer question 6& 7. 6.) What method/rules are used to find the limit when x ∞ ? 7.) Use the rules for finding a limit as x approaches infinity to find each limit a-c. a.) = b.) = c.) = The Limit from the left, the limit from the right:Left and Right hand limits and the General limit. To find a left hand limit means to find the limit as the limiting number is approached only from the left of the limiting number. To find a right hand limit means to find the limit as the limiting number is approached only from the right of the limiting number. The limit of a function can be different from the left than it is from the right. But if the limit from the left equals the same value as the limit from the right, then the general limit exists and is that value. -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 8.) Using the table below, answer the questions about x -.01 - .001 -.0001 .01 .001 .0001 f(x) 2.1 2.0001 2.0000001 .9 .9999 .9999999 Part II: (a) Piecewise functions (b) discontinuities (a) A piecewise function is defined using two or more equations each with a specific domain. Each equation gives a part of the graph based on your choice of x. (b) A function is considered to be discontinuous at a point, a, if any of the following are true: 1.) f(c) is undefined 2.) any small change in x (a move to the left or right on the x-axis) produces a “large” change in f(x) 3.) does not exist or is undefined. A discontinuity at x = a is removable if exists. Given the following piecewise function and it’s graph below answer questions 9 – 16. 9.) For x = -2, What is the right hand limit ? 10.) Does exist ? Why or Why Not ? If so, find 11.) Does this function have a discontinuity at x = -2 ? If so, name the reason why. Is it a removable discontinuity ? Why or why not 12.) 13.) = 14.) What feature can be found in the graph at x = 4 ? 15.) for x > 4 , find 16.) Using interval notation describe where this graph is continuous. 17.) Is it possible for a function to have a limit from the left be infinite and the limit from the right be zero if both limits are approaching the same number ? Sketch a graph if possible. 18.) Considering that discontinuities occur at holes, jumps, and vertical asymptotes. Is it possible for a function to have a limit from the left of ∞and the limit from the right -∞ if the number that is being approached is NOT the location of a vertical asymptote ? Explain. Part III: Derivatives: gives us the rate of change of what a function is modelling. If the function is describing the position of a moving object, the derivative can tell us more about how the object is moving, like the velocity or acceleration of the object. An application of the derivative: Position and Velocity The position function is a measure of location of an object along a path of travel with respect to time. As the object moves along its path, it goes through many changes as time elapses like changes in position and velocity. Velocity is a measure of how fast and can also indicate if the object is speeding up, slowing down, or at a constant. Velocity has two measures, average velocity and instantaneous velocity. To find Average velocityuse the position function, s(t), and a time period t1 to t2. Average velocity is the change in position over a period of time. , t1< t2 Instantaneous velocity is how fast a particle is going at a particular instant in time. To find it, you must use the first derivative of the position function. v(t) = , note: h is the same as ∆x The Acceleration is found by taking the derivative of the velocity function. a(t) = 19.) Given that a moving object’s path follows the function a.) Use the definition v(t) = to find the expression for the Velocity of the object where the initial velocity is 50ft/sec and the initial height is 600ft. Show all work. b.) Find the average velocity of the object between t = 1 and t = 3 seconds. c.) Find the instantaneous velocity of the object at t = 2 seconds A graph and its derivative in general: For any curve f(x), f ’(x) gives the rate of change of the y’s with respect to the x’s on the graph of the function, also known as the slope of the curve. Like a straight line, a curve has a positive slope if it is going uphill and a negative slope if it is going downhill (from left to right on the Cartesian Plane). 20.) Plot the function and it’s derivative in your graphing calculator. Look carefully at the two graphs: How does the graph of f’(x) reflect the changes in the slope of the graph of the curve f(x) ? 21.) In a parallel circuit, current moves through multiple paths. That means two resistors that are wired in parallel circuits havea lower total resistance than either of the parallel resistors. So, if you know what the smaller resistor is, then as the other resistor gets larger, the total resistance will never be larger than the smallest resistance in the circuit.It has a limit. The formula for resulting resistance, RT,(the combined resistance) of two resistors in a parallel circuit, one R1and the other R2 is . a.) If R1 is 10 Ω, find the limit of RT as R2  ∞. b.) Find RT for R1 = 10 and R2 = 5 c.) Find RT for R1 = 10 and R2 = 250 d.) Given b.) and c.) above, will RT ever go above 10 ? Explain why. 22.) Give the formula for the slope of a straight line and then the formula for the slope of a curve. How are the formulas the alike ? How are they different ?