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Questions and Answers
What is the slope of the normal at the point with abscissa x = -2 for the function ƒ(x) = |x^2 - |x||?
What is the slope of the normal at the point with abscissa x = -2 for the function ƒ(x) = |x^2 - |x||?
If y = 4x - 5 is a tangent to the curve y^2 = px^3 + q at the point (2, 3), what are the values of p and q?
If y = 4x - 5 is a tangent to the curve y^2 = px^3 + q at the point (2, 3), what are the values of p and q?
For the function ƒ(x) = |x^2 - |x||, what is the form of the function at x = -2?
For the function ƒ(x) = |x^2 - |x||, what is the form of the function at x = -2?
What is the derivative of y^2 = px^3 + q evaluated at the point (2, 3)?
What is the derivative of y^2 = px^3 + q evaluated at the point (2, 3)?
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In the function y = 4x - 5, which describes its role concerning the curve y^2 = px^3 + q?
In the function y = 4x - 5, which describes its role concerning the curve y^2 = px^3 + q?
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What does the slope of 6 correspond to in the context of the derivative for the curve y^2 = px^3 + q?
What does the slope of 6 correspond to in the context of the derivative for the curve y^2 = px^3 + q?
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If the slope of the tangent line is found to be 2p = 4, what is the value of p?
If the slope of the tangent line is found to be 2p = 4, what is the value of p?
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Which equation represents the slope of the tangent at the point where y^2 = px^3 + q and intersects y = 4x - 5?
Which equation represents the slope of the tangent at the point where y^2 = px^3 + q and intersects y = 4x - 5?
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What is the equation of the tangent to the curve ay = x at the point (at, at)?
What is the equation of the tangent to the curve ay = x at the point (at, at)?
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Which of the following represents the equation of the normal to the curve y = x^3 – 2x^2 + 4 at the point x = 2?
Which of the following represents the equation of the normal to the curve y = x^3 – 2x^2 + 4 at the point x = 2?
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What is the slope of the normal to the curve defined by x = a(q – sinq) and y = a(1 – cosq) at the point q = π/2?
What is the slope of the normal to the curve defined by x = a(q – sinq) and y = a(1 – cosq) at the point q = π/2?
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At which point does the tangent line to the curve y = x^3 – 2x^2 + 4 have a slope of 0?
At which point does the tangent line to the curve y = x^3 – 2x^2 + 4 have a slope of 0?
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What is the correct interpretation of the tangent at the point where x = at on the curve defined by ay = x?
What is the correct interpretation of the tangent at the point where x = at on the curve defined by ay = x?
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Which equation correctly describes the interaction between the curve y = x^3 – 2x^2 + 4 and its normal line?
Which equation correctly describes the interaction between the curve y = x^3 – 2x^2 + 4 and its normal line?
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What does the notation at represent in the context of tangents and normals on the curve ay = x?
What does the notation at represent in the context of tangents and normals on the curve ay = x?
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What characterizes the equation of the normal line compared to the tangent line?
What characterizes the equation of the normal line compared to the tangent line?
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What is the angle of intersection between the curves $y^2 = 8x$ and $x^2 = 4y - 12$ at the point $(2, 4)$?
What is the angle of intersection between the curves $y^2 = 8x$ and $x^2 = 4y - 12$ at the point $(2, 4)$?
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What is the length of the subtangent to the curve $x^2 + xy + y^2 = 7$ at the point $(1, -3)$?
What is the length of the subtangent to the curve $x^2 + xy + y^2 = 7$ at the point $(1, -3)$?
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How is the ratio of the square of the length of the normal to the square of the length of the tangent defined in relation to subtangent and subnormal?
How is the ratio of the square of the length of the normal to the square of the length of the tangent defined in relation to subtangent and subnormal?
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At what coordinates does the first curve $y^2 = 8x$ intersect with the second curve $x^2 = 4y - 12$?
At what coordinates does the first curve $y^2 = 8x$ intersect with the second curve $x^2 = 4y - 12$?
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If the length of the normal of a curve is given as $n$ and the length of the tangent as $t$, which of the following is true?
If the length of the normal of a curve is given as $n$ and the length of the tangent as $t$, which of the following is true?
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Which of the following statements correctly describes the characteristics of the curve $x^2 + xy + y^2 = 7$?
Which of the following statements correctly describes the characteristics of the curve $x^2 + xy + y^2 = 7$?
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Which angle represents the condition under which two curves are tangent to each other at their point of intersection?
Which angle represents the condition under which two curves are tangent to each other at their point of intersection?
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Which option describes the geometric significance of the subtangent to a curve?
Which option describes the geometric significance of the subtangent to a curve?
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At which point does the normal to the function y = f(x) at x = p pass through when the function is expressed as y = 1 - sin x?
At which point does the normal to the function y = f(x) at x = p pass through when the function is expressed as y = 1 - sin x?
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Which of the following points is NOT on the normal line to y = 1 - sin x at x = p?
Which of the following points is NOT on the normal line to y = 1 - sin x at x = p?
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If the normal to y = f(x) at x = p intersects the y-axis, what is the significance of the point (2, 2)?
If the normal to y = f(x) at x = p intersects the y-axis, what is the significance of the point (2, 2)?
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Which of the following points could be incorrectly chosen as a point on the normal line to y = f(x) at x = p?
Which of the following points could be incorrectly chosen as a point on the normal line to y = f(x) at x = p?
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Which of the following pairs of (x, y) could represent the point through which the normal to y = f(x) passes at x = p?
Which of the following pairs of (x, y) could represent the point through which the normal to y = f(x) passes at x = p?
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What condition must be satisfied for the curve y = ax^3 + bx^2 + cx + 5 to touch the x-axis at the point P (-2, 0)?
What condition must be satisfied for the curve y = ax^3 + bx^2 + cx + 5 to touch the x-axis at the point P (-2, 0)?
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If the gradient of the curve at the y-axis point Q is 3, what expression represents the derivative of the curve at that point?
If the gradient of the curve at the y-axis point Q is 3, what expression represents the derivative of the curve at that point?
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What is the correct format for the expression of the subnormal at any point on the curve xy = a n^n + 1?
What is the correct format for the expression of the subnormal at any point on the curve xy = a n^n + 1?
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In the curve defined by y = a ln(x^2 - a^2), how does the sum of the lengths of tangent and subtangent behave?
In the curve defined by y = a ln(x^2 - a^2), how does the sum of the lengths of tangent and subtangent behave?
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Which condition helps determine if the curves x^2/a^2 + y^2/b^2 = 1 and x^2/K1 + y^2/K2 = 1 intersect orthogonally?
Which condition helps determine if the curves x^2/a^2 + y^2/b^2 = 1 and x^2/K1 + y^2/K2 = 1 intersect orthogonally?
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For the two curves C1: x = y^2 and C2: xy = k, what value of k results in the curves cutting at right angles?
For the two curves C1: x = y^2 and C2: xy = k, what value of k results in the curves cutting at right angles?
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What is the implication of a gradient of 3 at the y-axis for the curve's coefficients?
What is the implication of a gradient of 3 at the y-axis for the curve's coefficients?
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In the equation of the curve y = ax^3 + bx^2 + cx + 5, what must be true about the coefficients a, b, and c given information about the point of tangency?
In the equation of the curve y = ax^3 + bx^2 + cx + 5, what must be true about the coefficients a, b, and c given information about the point of tangency?
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Study Notes
Normal Slope Calculation
- At ( x = -2 ), ( f(x) = |x^2 - |x|| ) simplifies to ( f(x) = x^2 + x ).
- The derivative ( \frac{dy}{dx} ) is calculated as ( 2x + 1 ), yielding a slope of -3 at ( x = -2 ).
- The slope of the normal line, which is perpendicular to the tangent, is given by ( \frac{1}{3} ).
Tangent to Curve
- The line ( y = 4x - 5 ) is tangent to the curve ( y^2 = px^3 + q ) at the point (2, 3).
- The derivative is represented as ( \frac{dy}{dx} = 8p + q ), and at ( x = 2 ), it equals 4.
- Substituting ( y = 3 ) yields the equation ( 6 = 3p(4) ), leading to ( p = 2 ) and ( q = -7 ).
Exercise Problems
- Multiple-choice questions come from tangents, normals, and slopes of curves:
- Determine the equation of the tangent for the curve defined as ( ay = x ).
- For the curve ( y = x^3 - 2x^2 + 4 ) at ( x = 2 ), find the equation of the normal line.
- Find the slope of the normal for the parametric curves defined by ( x = a(q - \sin q) ) and ( y = a(1 - \cos q) ) at ( q = \frac{\pi}{2} ).
Intersection of Curves
- The angle of intersection between the curves ( y^2 = 8x ) and ( x^2 = 4y - 12 ) is to be determined.
Subtangent Length
- The length of the subtangent for the curve ( x^2 + xy + y^2 = 7 ) at the point (1, -3) must be calculated.
Relationship of Tangent and Normal Lengths
- An important relationship is established, where the square of the length of the normal equals the square of the length of the tangent divided by the subnormal.
Specific Curve Behavior
- For the polynomial ( y = ax^3 + bx^2 + cx + 5 ), conditions regarding its interaction with the x-axis and y-axis gradients yield system constraints for ( a, b, c ).
- Evaluate the gradient of the line through (2,8) that is tangent to the curve ( y = x^3 ).
Constant Subnormal
- For the curve defined as ( xy = a^n ), determine the necessary value of ( n ) for consistent subnormal length across points.
Orthogonal Intersection of Curves
- Demonstrate that the curves ( \frac{x^2}{a^2} + \frac{y^2}{K_1} = 1 ) and ( \frac{x^2}{a^2} + \frac{y^2}{K_2} = 1 ) intersect orthogonally if ( K_1 \neq K_2 ).
- Calculate the value of ( k ) for the orthogonal intersection of curves ( C_1: x = y^2 ) and ( C_2: xy = k ).
Normal Line to Function
- Identify the point through which a normal to the function ( y = f(x) ) passes given the form ( \left( \frac{p}{4}, 0 \right) ) and similar variations.
Studying That Suits You
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Description
Test your understanding of tangent and normal slopes in calculus with this quiz. The questions cover various functions and their derivatives, focusing on calculating slopes and determining equations for tangents and normals. Perfect for students looking to solidify their knowledge in differential calculus.
