Podcast
Questions and Answers
What is the equation of the tangent to the curve $2(1 + y) = x^2$ at the origin, if it exists?
What is the equation of the tangent to the curve $2(1 + y) = x^2$ at the origin, if it exists?
- $y = 2x$
- $y = -2x$
- $y = x$
- $y = -x$ (correct)
At which point does the curve $2(1 + y) = x^2$ intersect the y-axis?
At which point does the curve $2(1 + y) = x^2$ intersect the y-axis?
- (0, -1) (correct)
- (0, 1)
- (0, 0)
- (0, 2)
What is the gradient of the curve $2(1 + y) = x^2$ at the origin?
What is the gradient of the curve $2(1 + y) = x^2$ at the origin?
- $0$
- $1$
- $-1$ (correct)
- $2$
Which of the following represents the correct transformation of the curve $2(1 + y) = x^2$ into standard form?
Which of the following represents the correct transformation of the curve $2(1 + y) = x^2$ into standard form?
What is the nature of the curve given by the equation $2(1 + y) = x^2$?
What is the nature of the curve given by the equation $2(1 + y) = x^2$?
What is the form of the equation representing the tangent line to the curve at the origin?
What is the form of the equation representing the tangent line to the curve at the origin?
What is the necessary condition for the tangent to exist at the origin for the curve?
What is the necessary condition for the tangent to exist at the origin for the curve?
What can be inferred if the derivative of the curve at the origin is zero?
What can be inferred if the derivative of the curve at the origin is zero?
Which of the following characterizes the slope of the tangent line to the curve at the origin if it exists?
Which of the following characterizes the slope of the tangent line to the curve at the origin if it exists?
What type of geometric figure does the equation $2(1 + y) = x^2$ represent?
What type of geometric figure does the equation $2(1 + y) = x^2$ represent?
Flashcards
Tangent to a Curve
Tangent to a Curve
The equation of a line that touches a curve at a single point, and whose slope is the same as the curve's derivative at that point.
Origin
Origin
The point where the x and y coordinates are both zero. Represented as (0,0).
Slope of a Curve
Slope of a Curve
The rate of change of a function at a specific point. Found by calculating the derivative of the function.
Equation of a Curve
Equation of a Curve
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Finding the Derivative
Finding the Derivative
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Study Notes
- The given equation is 2(1 + y) = x2 + x.
- To find the equation of the tangent at the origin (0,0), we need to determine if the origin is a point on the curve.
- Substituting x = 0 and y = 0 into the equation, we get 2(1 + 0) = 02 + 0, which simplifies to 2 = 0. This is not true, so the origin is not on the curve.
- Therefore, there is no tangent to the curve at the origin.
- The text mentions "the equation of the tangent to the curve..." implying a different goal, potentially finding the tangent at a different point.
- The existence of a tangent at the origin depends crucial on whether the origin lies on the curve.
- If the origin was on the curve, various methods to find the tangent would be possible, such as implicit differentiation or finding a point nearby.
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Description
This quiz explores the concept of tangents to curves through a specific equation. You will learn how to determine if points are on the curve and analyze the implications of these findings. Test your understanding of these fundamental concepts in calculus.
