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Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is
- a)y = 3x -5
- b)y = 3x +5
- c)3y = x+15
- d)3y = x -15
Correct answer is option 'B'. Can you explain this answer?
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Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is...
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Equation of the line normal to function f(x) = (x-8)^2/3 + 1 at P(0,5)
To find the equation of the line normal to the function at point P(0,5), we need to find the slope of the tangent line to the function at point P and then determine the negative reciprocal of that slope to get the slope of the normal line.
Finding the slope of the tangent line:
To find the slope of the tangent line, we need to find the derivative of the function f(x).
Given: f(x) = (x-8)^2/3 + 1
Taking the derivative of f(x) with respect to x will give us the slope of the tangent line at any given point.
f'(x) = (2/3) * (x-8)^(-1/3)
Now, substitute x = 0 into the derivative to find the slope of the tangent line at point P.
f'(0) = (2/3) * (0-8)^(-1/3)
= (2/3) * (-8)^(-1/3)
= (2/3) * (-1/2)
= -1/3
So, the slope of the tangent line to the function f(x) at point P(0,5) is -1/3.
Finding the slope of the normal line:
The slope of the normal line is the negative reciprocal of the slope of the tangent line. So, the slope of the normal line is 3.
Writing the equation of the normal line:
We have the slope of the normal line (m = 3) and the point it passes through (P(0,5)). We can use the point-slope form of a line to write the equation of the normal line.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 5 = 3(x - 0)
y - 5 = 3x
y = 3x + 5
The equation of the line normal to the function f(x) = (x-8)^2/3 + 1 at point P(0,5) is y = 3x + 5.
Therefore, the correct answer is option B) y = 3x + 5.
To find the equation of the line normal to the function at point P(0,5), we need to find the slope of the tangent line to the function at point P and then determine the negative reciprocal of that slope to get the slope of the normal line.
Finding the slope of the tangent line:
To find the slope of the tangent line, we need to find the derivative of the function f(x).
Given: f(x) = (x-8)^2/3 + 1
Taking the derivative of f(x) with respect to x will give us the slope of the tangent line at any given point.
f'(x) = (2/3) * (x-8)^(-1/3)
Now, substitute x = 0 into the derivative to find the slope of the tangent line at point P.
f'(0) = (2/3) * (0-8)^(-1/3)
= (2/3) * (-8)^(-1/3)
= (2/3) * (-1/2)
= -1/3
So, the slope of the tangent line to the function f(x) at point P(0,5) is -1/3.
Finding the slope of the normal line:
The slope of the normal line is the negative reciprocal of the slope of the tangent line. So, the slope of the normal line is 3.
Writing the equation of the normal line:
We have the slope of the normal line (m = 3) and the point it passes through (P(0,5)). We can use the point-slope form of a line to write the equation of the normal line.
The point-slope form of a line is given by:
y - y1 = m(x - x1)
Plugging in the values, we get:
y - 5 = 3(x - 0)
y - 5 = 3x
y = 3x + 5
The equation of the line normal to the function f(x) = (x-8)^2/3 + 1 at point P(0,5) is y = 3x + 5.
Therefore, the correct answer is option B) y = 3x + 5.
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Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer?.
Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Equation of the line normal to function f(x) = (x-8)2/3+1 at P(0,5) is a)y = 3x -5 b)y = 3x +5 c)3y = x+15 d)3y = x -15 Correct answer is option 'B'. Can you explain this answer?.
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