Matéria de Introdução Aos Cálculos responder em Inglês Enviar 9 1 9 2 9 3

Assignment 92 1 State in your own words what it means for a function f to be continuous at xc 2 State in your own words what it means for a function to be continuous on the interval ab 3 For the following exercises determine why the function f is discontinuous at a given point a on the graph State which condition fails 4 For the following exercises determine whether or not the given function f is continuous everywhere If it is continuous everywhere it is defined state for what range it is continuous If it is discontinuous state where it is discontinuous 5 Determine the values of b and c such that the following function is continuous on the entire real number line 6 Each square represents one square unit For each value of a determine which of the three conditions of continuity are satisfied at xa and which are not a x 3 b x 2 c x 4 Assignment 93 1 How is the slope of a linear function similar to the derivative 2 What is the difference between the average rate of change of a function on the interval x xh and the derivative of the function at x 3 A car traveled 110 miles during the time period from 200 PM to 400 PM What was the cars average velocity At exactly 230 PM the speed of the car registered exactly 62 miles per hour What is another name for the speed of the car at 230 PM Why does this speed differ from the average velocity 4 Explain the concept of the slope of a curve at point x 5 Suppose water is flowing into a tank at an average rate of 45 gallons per minute Translate this statement into the language of mathematics 6 For the following exercises use the definition of derivative to calculate the derivative of each function 7 For the following exercises find the average rate of change between the two points 8 For the following polynomial functions find the derivatives 9 For the following functions find the equation of the tangent line to the curve at the given point x on the curve 10 For the following exercises explain the notation in words The volume ft of a tank of gasoline in gallons t minutes after noon Assignment 91 1 Explain the difference between a value at xa and the limit as x approaches a 2 Explain why we say a function does not have a limit as x approaches a if as x approaches a the lefthand limit is not equal to the righthand limit 3 For the following exercises estimate the functional values and the limits from the graph of the function f provided in the graph below 4 For the following exercises draw the graph of a function from the functional values and limits provided a b c 5 For the following exercises use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as x approaches a If the function has a limit as x approaches a state it If not discuss why there is no limit 6 For the following exercises evaluate the limits algebraically lim x0 3 lim x2 5x x2 1 lim x2 x2 5x 6 x 2 lim x3 x2 9 x 3 lim x1 x2 2x 3 x 1 lim x32 6x2 17x 12 2x 3 lim x72 8x2 18x 35 2x 7 Assignment 91 1 Explain the difference between a value at xa and the limit as x approaches a Value at xa This refers to the actual value of the function fa meaning the point where the function is defined at xa Limit as x approaches a This represents the value that the function approaches as x gets closer to a from both sides left and right Its possible that the limit exists even if fa is undefined or the limit may differ from the actual function value at xa 2 Explain why we say a function does not have a limit as x approaches a if as x approaches a the lefthand limit is not equal to the righthand limit A function does not have a limit as x approaches a when the limit from the left side 𝑙𝑖𝑚𝑥𝑎𝑓𝑥 is not equal to the limit from the right side In such cases the function approaches 𝑙𝑖𝑚𝑥𝑎𝑓𝑥 different values from the left and right meaning no single value is approached as x gets closer to aaa 3 For the following exercises estimate the functional values and the limits from the graph of the function f provided in the graph below 𝑙𝑖𝑚𝑥2 𝑓𝑥 3 𝑙𝑖𝑚𝑥2 𝑓𝑥 3 𝑙𝑖𝑚𝑥2𝑓𝑥 3 𝑓 2 3 𝑙𝑖𝑚𝑥1 𝑓𝑥 3 𝑙𝑖𝑚𝑥1 𝑓𝑥 𝑑𝑜𝑒𝑠𝑛𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑖𝑚𝑥1𝑓𝑥 𝑑𝑜𝑒𝑠𝑛𝑡 𝑒𝑥𝑖𝑠𝑡 𝑓1 𝑑𝑜𝑒𝑠𝑛𝑡 𝑒𝑥𝑖𝑠𝑡 𝑙𝑖𝑚𝑥4 𝑓𝑥 5 𝑙𝑖𝑚𝑥4 𝑓𝑥 4 4 For the following exercises draw the graph of a function from the functional values and limits provided a b c 5 For the following exercises use a graphing utility to find numerical or graphical evidence to determine the left and righthand limits of the function given as x approaches a If the function has a limit as x approaches a state it If not discuss why there is no limit lim x0 e1 x² D C lim x0 x x A B 6 For the following exercises evaluate the limits algebraically 𝑙𝑖𝑚𝑥03 The function is constant so the limit is just 3 𝑙𝑖𝑚𝑥2 5𝑥 𝑥 21 52 2 21 10 41 10 3 𝑙𝑖𝑚𝑥2 𝑥 25𝑥6 𝑥2 𝑥 25𝑥6 𝑥2 𝑥2𝑥3 𝑥2 2223 22 01 4 0 4 0 𝑙𝑖𝑚𝑥3 𝑥 29 𝑥3 𝑥3𝑥3 𝑥3 𝑥 3 3 3 6 𝑙𝑖𝑚𝑥1 𝑥 22𝑥3 𝑥1 𝑥3𝑥1 𝑥1 𝑥 3 1 3 4 𝑙𝑖𝑚 3 2 6𝑥 217𝑥12 2𝑥3 2𝑥33𝑥4 2𝑥3 3𝑥 4 9 2 4 1 2 𝑙𝑖𝑚𝑥 7 2 8𝑥 218𝑥35 2𝑥7 8 7 2 218 7 2 35 2 7 2 7 986335 77 Assignment 92 1 State in your own words what it means for a function f to be continuous at xc For a function f to be continuous at xc it means that there are no sudden jumps breaks or holes in the graph at that point In more technical terms three things must happen 1 fc exists meaning the function has a value at xc 2 The limit of fx as x approaches c exists 3 The limit of fx as x approaches c is equal to the actual value fc 2 State in your own words what it means for a function to be continuous on the interval ab For a function to be continuous on the interval ab it means the function behaves smoothly throughout that entire range of values Specifically at every point between a and b the function has no breaks jumps or holes and you can draw the graph without lifting your pen 3 For the following exercises determine why the function f is discontinuous at a given point a on the graph State which condition fails 4 For the following exercises determine whether or not the given function f is continuous everywhere If it is continuous everywhere it is defined state for what range it is continuous If it is discontinuous state where it is discontinuous The sine function is continuous everywhere so this function is continuous for all real numbers Conclusion The function is continuous everywhere Polynomials are continuous everywhere so this function is continuous for all real numbers Conclusion The function is continuous everywhere This is a rational function so it is continuous everywhere except where the denominator is zero The denominator is zero at x5 Conclusion The function is discontinuous at x5 Exponential functions are continuous everywhere Conclusion The function is continuous everywhere The denominator is which is zero at x0 and x2 𝑥 2 2𝑥 𝑥𝑥 2 Conclusion The function is discontinuous at x0 and x2 The tangent function is undefined where for integer n because at these points 𝑥 π 2 𝑛π the tangent function has vertical asymptotes Conclusion The function is discontinuous at 𝑥 π 2 𝑛π The function has a term 5 which is undefined at x0 𝑥 Conclusion The function is discontinuous at x0 5 Determine the values of b and c such that the following function is continuous on the entire real number line For continuity at x 3 the two pieces of the function must be equal at this point as well That is lim x3 x 1 lim x3 x² bx c Evaluating both sides Left side x 1 at x 3 3 1 4 Right side x² bx c at x 3 3² 3b c 9 3b c So we need 9 3b c 4 This simplifies to 3b c 5 Now we solve the system of equations b c 1 1 3b c 5 2 Subtract equation 1 from equation 2 3b c b c 5 1 2b 6 b 3 Substitute b 3 into equation 1 3 c 1 c 4 Thus the values of b and c that make fx continuous are b 3 c 4 6 Each square represents one square unit For each value of a determine which of the three conditions of continuity are satisfied at xa and which are not a x 3 From the graph there is a solid point at x3 meaning that f3 is defined Conclusion The first condition is satisfied As x approaches 3 from both sides the graph approaches the same value This suggests that the limit exists Conclusion The second condition is satisfied Since the limit equals the function value at x3 this condition is satisfied Conclusion The third condition is satisfied b x 2 At x2 there is a hole in the graph meaning that f2 is not defined Conclusion The first condition is not satisfied The graph approaches the same value from both the left and right as x approaches 2 meaning that the limit exists Conclusion The second condition is satisfied Since f2 does not exist this condition is not applicable Conclusion The third condition is not satisfied because f2 does not exist c x 4 At x4 there is a point defined below the graph an isolated dot so f4 exists at that specific value Conclusion The first condition is satisfied As x approaches 4 from both sides the graph approaches a different value than f4 meaning the limit does not approach the value of the function at x4 Conclusion The second condition is not satisfied because the graph jumps to a different point Since the limit does not exist in the same sense as the function value at x4 this condition is not satisfied Conclusion The third condition is not satisfied Assignment 93 1 How is the slope of a linear function similar to the derivative A linear function is of the form where m is the slope The slope of a 𝑓𝑥 𝑚𝑥 𝑏 linear function is constant and represents the rate of change of the function The derivative represents the instantaneous rate of change of a function at a specific 𝑓𝑥 point For a linear function the derivative is constant and equals the slope 𝑓𝑥 𝑚 So a linear function the derivative and the slope are the same Both indicate a constant rate of change 2 What is the difference between the average rate of change of a function on the interval x xh and the derivative of the function at x The average rate of change of a function over an interval is 𝑓𝑥 𝑥 𝑥 ℎ 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑓𝑥ℎ𝑓𝑥 ℎ This gives the rate of change over the entire interval The derivative of the function at 𝑥 is the limit of the average rate of change as 𝑓𝑥 ℎ 0 𝑓𝑥 𝑙𝑖𝑚ℎ0 𝑓𝑥ℎ𝑓𝑥 ℎ The average rate of change considers the functions change over a finite interval while the derivative gives the instantaneous rate of change at a specific point 3 A car traveled 110 miles during the time period from 200 PM to 400 PM What was the cars average velocity At exactly 230 PM the speed of the car registered exactly 62 miles per hour What is another name for the speed of the car at 230 PM Why does this speed differ from the average velocity 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑇𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 110 𝑚𝑖𝑙𝑒𝑠 2 ℎ𝑜𝑢𝑟𝑠 55 𝑚𝑖𝑙𝑒𝑠 𝑝𝑒𝑟 ℎ𝑜𝑢𝑟 4 Explain the concept of the slope of a curve at point x Instantaneous velocity measures the cars speed at a single moment whereas the average velocity is calculated over a time period thus they can differ 5 Suppose water is flowing into a tank at an average rate of 45 gallons per minute Translate this statement into the language of mathematics The slope of a curve at a point x is the slope of the tangent line to the curve at that point The slope of the tangent line represents the instantaneous rate of change of the function at x which is the derivative of the function at that point Mathematically if the function is fx the slope at xa is 𝑥 𝑎 𝑓𝑎 This slope tells us how the function behaves increasing or decreasing at the point xa 6 For the following exercises use the definition of derivative to calculate the derivative of each function 𝑓𝑥 3𝑥 4 𝑓𝑥 ℎ 3𝑥 ℎ 4 3𝑥 3ℎ 4 3𝑥3ℎ43𝑥4 ℎ 3ℎ ℎ 3 𝑓𝑥 2𝑥 1 𝑓𝑥 ℎ 2𝑥 ℎ 1 2𝑥 2ℎ 1 2𝑥2ℎ12𝑥1 ℎ 2ℎ ℎ 2 𝑓𝑥 𝑥 2 2𝑥 1 𝑓𝑥 ℎ 𝑥 ℎ 2 2𝑥 ℎ 1 𝑥 2 2𝑥ℎ ℎ 2 2𝑥 2ℎ 1 𝑥 22𝑥ℎℎ 22𝑥2ℎ1𝑥 22𝑥1 ℎ 2𝑥ℎℎ 22ℎ ℎ 2𝑥 ℎ 2 7 For the following exercises find the average rate of change between the two points 2 0 𝑎𝑛𝑑 4 5 3 2 5 4 3 𝑎𝑛𝑑 2 1 1 2 0 5 𝑎𝑛𝑑 6 5 3 5 7 2 𝑎𝑛𝑑 7 10 7 4 8 For the following polynomial functions find the derivatives 𝑓𝑥 𝑥 3 1 3𝑥 2 𝑓𝑥 3𝑥 2 7𝑥 6𝑥 7 𝑓𝑥 7𝑥 2 14𝑥 𝑓𝑥 3𝑥 3 2𝑥 2 𝑥 26 9𝑥 2 4𝑥 1 9 For the following functions find the equation of the tangent line to the curve at the given point x on the curve 𝑓𝑥 2𝑥 2 3𝑥 𝑥 3 𝑓𝑥 4𝑥 3 𝑓3 43 3 9 𝑓3 23 2 33 29 9 9 𝑦 9 9𝑥 3 𝑦 9𝑥 27 9 9𝑥 18 𝑓𝑥 𝑥 3 1 𝑥 2 𝑓𝑥 3𝑥 2 𝑓2 32 2 34 12 𝑓2 2 3 1 8 1 9 𝑦 9 12𝑥 2 𝑦 12𝑥 24 9 12𝑥 15 𝑓𝑥 𝑥 𝑥 9 𝑓𝑥 1 2 𝑥 12 𝑓9 1 2 9 12 1 2 9 1 23 1 6 𝑓9 9 3 𝑦 3 1 6 𝑥 9 𝑦 1 6 𝑥 3 2 3 10 For the following exercises explain the notation in words The volume ft of a tank of gasoline in gallons t minutes after noon At 0000 the volume was 600 gallons 𝑓0 600 At 0030 the volume had reduced 20 gallons each minute 𝑓30 20 At 0030 the volume was 0 gallons 𝑓30 0 At 0320 the volume had increased 30 gallons each minute 𝑓200 30 𝑓240 500 At 0400 the volume was 500 gallons