Section 11.4:

1. Select the correct choices that complete the sentences below. The slope of the tangent line to the graph of a function y= f(x) at (a,f(a)) is given by lim (1) provided that this limit exists. h-0 This limit also describes the (2) rate of change of (3) with respect to (4) at a (1) f(a + h) + f(a) f(a) - f(a + h) (2) O eventual (3) Ox (4) Of h O instantaneous Of O x f(a) - f(a + h) f(a + h) - f(a) O h ((a + h) + f(a) O O f(a + h) - f(a) 2. Fill in the blanks so that the resulting statement is true. f(1 + h) - f(1) Using f(x) = 3x + x, we can determine that lim =7. This means that the point-slope equation of the tangent line to the graph h-0 n of f(x) = 3x" + x at (1,4) is y - (x - Using f(x) = 3x"+ x, we can determine that lim f(1 + h) - f(1) - =7. This means that the point-slope equation of the tangent line to the graph h of f(x) = 3x + x at (1,4) is y - (1) = (2) (x - (3) (1) 0 4 (2) 0 4 (3) 0 7 0 7 O 0 1 0 1 0 4 3. Fill in the blanks so that the resulting statement is true. Using f(x) = x"- 9x + 5, we can determine that f'(x) =2x -9. This means that the point-slope equation of the tangent line to the graph of f(x) = x - 9x + 5 at (8,22) is y - _(X - _ Using f(x) = x" - 9x + 5, we can determine that f (x) = 2x -9. This means that the point-slope equation of the tangent line to the graph of f(x) = x -9x + 5 at (8,22) is y - (1) = (2) (x - (3) (1) 0 7 (2) 0 10 (3) 0 10 O 8 O 22 O B O 10 O 8 07 O 22 O7 O 224. a. Find the slope of the tangent line to the graph of f(x) =9x" at the point (2,36). b. Find the slope-intercept equation of the tangent line to the graph of f(x) =9x" at the point (2,36). a. What is the slope of the tangent line? mtan b. Complete the slope-intercept equation of the tangent line below. y = https://xlitemprod.pearsonomg.com/api/v1/print/highered 12/7/22, 7:47 PM Sec11.4-Adriana Lumbreras 5. a. Find the slope of the tangent line to the graph of f at the given point. b. Find the slope-intercept equation of the tangent line to the graph of f at the given point. f(x) =5x - 2x at (2,16) a. The slope is (Type an integer or a simplified fraction.) b. y= (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.) 6. a. Find the slope of the tangent line to the graph of f at the given point. b. Find the slope-intercept equation of the tangent line to the graph of f at the given point. (x) =413x at (3,12) a. The slope is (Type an integer or a simplified fraction.) b. y= (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)7. a. Find the derivative of f at x. That is, find f'(x). b. Find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function. (x) = - 5x + 5; x= -4, x =1 a. What is the derivative of f(x) = - 5x + 5 at x? f' (x) = (Simplify your answer.) b. What is the slope of the tangent line to f(x) = - 5x +5 at x = -4? mtan = (Simplify your answer.) What is the slope of the tangent line to f(x) = -5x + 5 at x = 1? mtan (Simplify your answer.) 8. a. Find the derivative of fat x. That is, find f'(x). b. Find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function. 25 f(X) = Fix= -5, x = 1 a. f'(x) = b. The slope of the tangent line at x = -5 is The slope of the tangent line at x = 1 is 9. The function f(x) = 6xx" describes the volume, f(x), of a right circular cylinder of height 6 feet and radius x feet. If the radius is changing, find the instantaneous rate of change of the volume with respect to the radius when the radius is 5 feet. The instantaneous rate of change of the volume is ( 1 ) . (Type an exact answer, using * as needed.) (1) O f/fts. Oft. Off /ft.10. Find the derivative of the function at the given number using a graphing utility. f(x)= x"- x+ x - x+ 2 at 1 f(x) = (Type an integer or a decimal rounded to two decimal places as needed. ) 11. Watch the video and then solve the problem given below. Click here to watch the video.1 a. Find the slope of the tangent line to the graph of f at the given point. b. Find the slope-intercept equation of the tangent line to the graph of f at the given point. f(x) = 414x at (4,16) a. The slope is . (Type an integer or a simplified fraction.) b. y= (Type your answer in slope-intercept form. Use integers or fractions for any numbers in the expression.)12. Watch the video and then solve the problem given below. Click here to watch the video. a. Find the derivative of f at x. That is, find f (x). b. Find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function. f(x) =x -9; x= -2, x= -1 a. f' (x) = (Simplify your answer. Use integers or fractions for any numbers in the expression.) b. At x = - 2, the slope of the tangent line to the graph of f is (Type an integer or a simplified fraction.) At x = - 1, the slope of the tangent line to the graph of f is (Type an integer or a simplified fraction.) 2: http://https://mediaplayer.pearsonomg.com/assets/bzpc6e_11_04_02 13. An explosion causes debris to rise vertically with an initial velocity of 112 feet per second. The function s(t) = - 161 + 112t describes the height of the debris above the ground, s(t). in feet, t seconds after the explosion. a. What is the instantaneous velocity of the debris 1 second after the explosion? 3 seconds after the explosion? b. What is the instantaneous velocity of the debris when it hits the ground? a. The instantaneous velocity of the debris 1 second after the explosion is feet per second. The instantaneous velocity of the debris 3 seconds after the explosion is feet per second. b. The instantaneous velocity of the debris when it hits the ground is feet per second.14. A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 83 feet per second. The function s(t) = - 161" + 83t+ 4 describes the ball's height above the ground, s(1), in feet, t seconds after it was hit. a. What is the instantaneous velocity of the ball 2 seconds after it was hit? 4 seconds after it was hit? b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? What is its maximum height? a. What is the instantaneous velocity of the ball 2 seconds after it was hit? V= feet per second What is the instantaneous velocity of the ball 4 seconds after it was hit? V = feet per second b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? 1= seconds Round to one decimal place as needed.) What is the maximum height the ball reaches? h =] feet 15. A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 122 feet per second. The function s(1) = - 161" + 122t +4 describes the ball's height above the ground, s(t), in feet, t seconds after it was hit. a. What is the instantaneous velocity of the ball 2 seconds after it was hit? 4 seconds after it was hit? b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? What is its maximum height? a. What is the instantaneous velocity of the ball 2 seconds after it was hit? V= feet per second What is the instantaneous velocity of the ball 4 seconds after it was hit? V= feet per second b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? 1= seconds (Round to one decimal place as needed.) What is the maximum height the ball reaches? feet15. An explosion causes debris to rise vertically with an initial velocity of 48 feet per second. The function s(t) = -161 + 48t describes the height of the debris above the ground, s(), in feet, t seconds after the explosion. a. What is the instantaneous velocity of the debris 1 second after the explosion? 2 seconds after the explosion? b. What is the instantaneous velocity of the debris when it hits the ground? a. The instantaneous velocity of the debris 1 second after the explosion is feet per second. The instantaneous velocity of the debris 2 seconds after the explosion is feet per second. b. The instantaneous velocity of the debris when it hits the ground is feet per second. https://xlitemprod.pearsoncing.com/api/v1/print/highered 4/5 12/7/22, 7:47 PM Sec11.4-Adriana Lumbreras 17. A foul tip of a baseball is hit straight upward from a height of 4 feet with an initial velocity of 87 feet per second. The function s(1) = - 161" + 87t+ 4 describes the ball's height above the ground, s(1), in feet, t seconds after it was hit. a. What is the instantaneous velocity of the ball 2 seconds after it was hit? 4 seconds after it was hit? b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? What is its maximum height? a. What is the instantaneous velocity of the ball 2 seconds after it was hit? V= feet per second What is the instantaneous velocity of the ball 4 seconds after it was hit? V = feet per second b. The ball reaches its maximum height above the ground when the instantaneous velocity is zero. After how many seconds does the ball reach its maximum height? t = seconds 'Round to one decimal place as needed.) What is the maximum height the ball reaches? h = feet