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question:Given the function h, left bracket, x, right bracket, equals, x, squared, minus, 10, x, plus, 21h(x)=x 2 −10x+21, determine the average rate of change of the function over the interval minus, 1, is less than or equal to, x, is less than or equal to, 10−1≤x≤10
answer:To determine the average rate of change of the function h(x) over the interval -1 ≤ x ≤ 10, we need to calculate the change in y-values divided by the change in x-values. First, let's find the y-values for x = -1 and x = 10 by substituting them into the function: For x = -1: h(-1) = (-1)^2 - 10(-1) + 21 = 1 + 10 + 21 = 32 For x = 10: h(10) = (10)^2 - 10(10) + 21 = 100 - 100 + 21 = 21 Now, we can calculate the average rate of change: Average rate of change = (h(10) - h(-1)) / (10 - (-1)) = (21 - 32) / (10 + 1) = -11 / 11 = -1 Therefore, the average rate of change of the function h(x) over the interval -1 ≤ x ≤ 10 is -1.
question:Using this example do the previous question again. Given the function f, left bracket, x, right bracket, equals, minus, x, squared, minus, 7, x, plus, 20f(x)=−x 2 −7x+20, determine the average rate of change of the function over the interval minus, 8, is less than or equal to, x, is less than or equal to, 0−8≤x≤0. Average Rate of Change: Average Rate of Change: start fraction, f, left bracket, x 2, right bracket, minus, f, left bracket, x 1, right bracket, divided by, x 2, minus, x 1, end fraction, or, start fraction, y 2, minus, y 1, divided by, x 2, minus, x 1, end fraction x 2 −x 1 f(x 2 )−f(x 1 ) or x 2 −x 1 y 2 −y 1 Equivalent formulas f, left bracket, x, right bracket, equals, minus, x, squared, minus, 7, x, plus, 20 f(x)=−x 2 −7x+20 x 1, equals, minus, 8x 1 =−8 x 2, equals, 0x 2 =0 f, left bracket, minus, 8, right bracketf(−8) f, left bracket, 0, right bracketf(0) minus, left bracket, minus, 8, right bracket, squared, minus, 7, left bracket, minus, 8, right bracket, plus, 20−(−8) 2 −7(−8)+20 minus, left bracket, 0, right bracket, squared, minus, 7, left bracket, 0, right bracket, plus, 20−(0) 2 −7(0)+20 1212 2020 Ordered Pairs: , left bracket, minus, 8, comma, 12, right bracket, and , left bracket, 0, comma, 20, right bracket Ordered Pairs: (−8,12) and (0,20) (x, f(x)) start fraction, f, left bracket, x 2, right bracket, minus, f, left bracket, x 1, right bracket, divided by, x 2, minus, x 1, end fraction, or, start fraction, y 2, minus, y 1, divided by, x 2, minus, x 1, end fraction x 2 −x 1 f(x 2 )−f(x 1 ) or x 2 −x 1 y 2 −y 1 start fraction, 20, minus, 12, divided by, 0, minus, left bracket, minus, 8, right bracket, end fraction 0−(−8) 20−12 Plug in start fraction, 8, divided by, 8, end fraction 8 8 Subtract 1 1 The average rate of change
answer:Using the given function f(x) = -x^2 - 7x + 20 and the interval -8 ≤ x ≤ 0, we can calculate the average rate of change. First, let's find the y-values for x = -8 and x = 0 by substituting them into the function: For x = -8: f(-8) = (-(-8))^2 - 7(-8) + 20 = 64 + 56 + 20 = 140 For x = 0: f(0) = -(0)^2 - 7(0) + 20 = 0 - 0 + 20 = 20 Now, we can calculate the average rate of change: Average rate of change = (f(0) - f(-8)) / (0 - (-8)) = (20 - 140) / (0 + 8) = (-120) / 8 = -15 Therefore, the average rate of change of the function f(x) over the interval -8 ≤ x ≤ 0 is -15.
question:If f, left bracket, x, right bracket, equals, x, cubed, plus, 2, x, squared, minus, 21, x, plus, 18f(x)=x 3 +2x 2 −21x+18 and f, left bracket, 3, right bracket, equals, 0f(3)=0, then find all of the zeros of f, left bracket, x, right bracketf(x) algebraically.
answer:To find the zeros of the function f(x) = x^3 + 2x^2 - 21x + 18 algebraically, we need to solve the equation f(x) = 0. Let's set f(x) equal to 0 and solve for x: x^3 + 2x^2 - 21x + 18 = 0 To solve this equation, we can try factoring, using the rational root theorem, or using numerical methods like graphing or using a calculator. Let's first try factoring the equation: (x - 1)(x + 3)(x - 6) = 0 Set each factor equal to zero: x - 1 = 0 x = 1 x + 3 = 0 x = -3 x - 6 = 0 x = 6 The zeros of the function f(x) are x = 1, x = -3, and x = 6. Therefore, the zeros of the function f(x) = x^3 + 2x^2 - 21x + 18 are x = 1, x = -3, and x = 6.
question:-7*-5
answer:-7 * -5 = 35