Calculus: Tangents and Normals Quiz

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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||?

$-3$ (correct)

$1$

$-1$

$3$

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?

p = -2, q = 7

p = 1, q = 0

p = -2, q = -7

p = 2, q = -7 (correct)

For the function ƒ(x) = |x^2 - |x||, what is the form of the function at x = -2?

ƒ(x) = -x^2 + x

ƒ(x) = x^2 - x

ƒ(x) = -x^2 - x

ƒ(x) = x^2 + x (correct)

What is the derivative of y^2 = px^3 + q evaluated at the point (2, 3)?

$8p + q$ Signup and view all the answers

In the function y = 4x - 5, which describes its role concerning the curve y^2 = px^3 + q?

It is a tangent to the curve at a point. Signup and view all the answers

What does the slope of 6 correspond to in the context of the derivative for the curve y^2 = px^3 + q?

It is the slope at the point (2, 3). Signup and view all the answers

If the slope of the tangent line is found to be 2p = 4, what is the value of p?

2 Signup and view all the answers

Which equation represents the slope of the tangent at the point where y^2 = px^3 + q and intersects y = 4x - 5?

9 = 8p + q Signup and view all the answers

What is the equation of the tangent to the curve ay = x at the point (at, at)?

3tx + 2y = at3 Signup and view all the answers

Which of the following represents the equation of the normal to the curve y = x^3 – 2x^2 + 4 at the point x = 2?

x + 4y = 18 Signup and view all the answers

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?

0 Signup and view all the answers

At which point does the tangent line to the curve y = x^3 – 2x^2 + 4 have a slope of 0?

x = 1 Signup and view all the answers

What is the correct interpretation of the tangent at the point where x = at on the curve defined by ay = x?

It is perpendicular to the curve. Signup and view all the answers

Which equation correctly describes the interaction between the curve y = x^3 – 2x^2 + 4 and its normal line?

The normal lies entirely above the curve. Signup and view all the answers

What does the notation at represent in the context of tangents and normals on the curve ay = x?

It is the coefficient of the variable x. Signup and view all the answers

What characterizes the equation of the normal line compared to the tangent line?

The normal line intersects the curve at the point of tangency. Signup and view all the answers

What is the angle of intersection between the curves $y^2 = 8x$ and $x^2 = 4y - 12$ at the point $(2, 4)$?

90° Signup and view all the answers

What is the length of the subtangent to the curve $x^2 + xy + y^2 = 7$ at the point $(1, -3)$?

3 Signup and view all the answers

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?

(subtangent)/(subnormal) Signup and view all the answers

At what coordinates does the first curve $y^2 = 8x$ intersect with the second curve $x^2 = 4y - 12$?

(2, 4) Signup and view all the answers

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?

$n^2 = t^2$ Signup and view all the answers

Which of the following statements correctly describes the characteristics of the curve $x^2 + xy + y^2 = 7$?

It is an ellipse. Signup and view all the answers

Which angle represents the condition under which two curves are tangent to each other at their point of intersection?

0° Signup and view all the answers

Which option describes the geometric significance of the subtangent to a curve?

Length from the point of tangency to the x-axis. Signup and view all the answers

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?

(1/4, 0) Signup and view all the answers

Which of the following points is NOT on the normal line to y = 1 - sin x at x = p?

(π, -1) Signup and view all the answers

If the normal to y = f(x) at x = p intersects the y-axis, what is the significance of the point (2, 2)?

It represents the y-intercept of the normal. Signup and view all the answers

Which of the following points could be incorrectly chosen as a point on the normal line to y = f(x) at x = p?

(1, 1) Signup and view all the answers

Which of the following pairs of (x, y) could represent the point through which the normal to y = f(x) passes at x = p?

(1/4, 0) Signup and view all the answers

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)?

The derivative must be zero at P Signup and view all the answers

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?

3 = 3a + 2b Signup and view all the answers

What is the correct format for the expression of the subnormal at any point on the curve xy = a n^n + 1?

It is always constant. Signup and view all the answers

In the curve defined by y = a ln(x^2 - a^2), how does the sum of the lengths of tangent and subtangent behave?

It varies as the product of the coordinates. Signup and view all the answers

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?

The derivatives at intersection points must be negative reciprocals. Signup and view all the answers

For the two curves C1: x = y^2 and C2: xy = k, what value of k results in the curves cutting at right angles?

k=1 Signup and view all the answers

What is the implication of a gradient of 3 at the y-axis for the curve's coefficients?

It indicates a specific relationship between a, b, and c. Signup and view all the answers

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?

The coefficients must satisfy a specific polynomial equation. Signup and view all the answers

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.

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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.