MATA33H3 Lecture 10: Lecture 10 (11.5 - 12.1)

A useful remark: Since the derivative function is defined through limits, it is easy to show the following result: If f(s) and g(x) are equal on an open interval (a, b), and f?(c) exists for some c E (a, b), then g?(c) also exists and g(c) = f (c). this allows one to calculate the derivative of functions Defined piecewise, except at the x values where the definition changes from one formula to another. For example: if then h (x) = 4x on (- infinity , 5) (because it agrees with (2x^2 + 4)? there) and h?(x) = 2e^2x on (5, infinity) (because it agrees with (e^2x) there). Whether or not h?(5) exists requires more work and we will see how to answer this in the following two questions. (1) Consider the function (a) Show that f(s) is continuous at x = 0 by calculating the left and right hand limits of f(s) as it approaches 0 and showing they are both equal to f(0). (b) Show that f(s) is not differentiable at 0 by calculating and showing the two sided limit does not exist (note: calculate these limits explicitly, do not take shortcuts using derivative formulas). These one - sided limits are called the left and right - sided derivatives of f(x) at 0. By definition, f?(0) exists if and only if the two one - sided derivatives both exists and are equal (note: there is nothing special about 0; one sided derivatives are defined similarity for any s value). (c) Using the Power Rule Theorem from class and the remark preceding this question, calculate f(x) (note that by (b), 0 should not be in the domain of .f?(x)). Sketch a graph of f(s) and f(x) on the same XY - axis (you do not need to be precise, just sketch time shape).

Math 1510 Preview A.14: Analysis Objective: There are many situations where functions arise as area functions for some continuous integrand, that is F(x)- ro) d In this activity you will use the tools of calculus to analyze such a function. Background: If the integrand f(t) has an elementary antiderivative then the second part of the Fundamental Theorem of Calculus gives a formula for the area function F(x). However, many functions do not have an elementary antiderivative. In such a situation we can use the first part of the Fundamental Theorem to get information about F through its derivative. You should review section 5.3 and make sure you understand both parts of the Fundamental Theorem. You will also want to use a CAS (for example Wolfram Alphal for some of your computations. CAUTION: Some systems (including Alpha) will report an antiderivative for the integral below but they are not useful in this problem because they incorporate assumptions that are not valid here. For calculating numeric estimates of area using the midpoint rule you may use the Desmos example The Midpoint Rule Steps: You will analyze the function 8 cos(e)+v 1. Domains. Describe the domains of the area function F(x) and the integrand() 2. Values. Use the midpoint rule to estimate the following values 4 3. Symmetry. Does P(x) have any properties such as symmetry leven or odd) or periodicity Page 1 of 2

Math 1510 Preview A.14: Analysis Objective There are many situations where functions arise as area functions for some continuous integrand, that is F(x)- 0) d In this activity you will use the tools of calculus to analyze such a function. Background If the integrand(r) has an elementary antiderivative then the second part of the Fundamentall Theorenm of Calculus gives a formula for the area function F(x). However, many functions do not have an elementary antiderivative. In such a situation we can use the first part of the Fundamental Theorem to get information about F through its derivative. You should review section 5.3 and make sure you understand both parts of the Fundamental Theorem. You will also want to use a CAS (for example Wolfram Alpha) for some of your computations. CAUTION: Some systems (including Alpha) will report an antiderivative for the integral below but they are not useful in this problem because they incorporate assumptions that are not valid here. For calculating numeric estimates of area using the midpoint rule you may use the Desmos example The Midpoint Rule Steps: You will analyze the function sin ()+1 1. Domains. Describe the domains of the area function F(x) and the integrand s). 2. Values. Use the midpoint rule to estimate the following values: 6 F(x) 3. Symmetry. Does F(x) have any properties such as symmetry (even or odd) or periodicity? Page 1 of 2