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What is the gradient of the curve y = x at the point x = -2?
What is the gradient of the curve y = x at the point x = -2?
-2
Explain how the gradient of the curve y = x changes as the value of x increases.
Explain how the gradient of the curve y = x changes as the value of x increases.
The gradient of the curve y = x increases linearly as the value of x increases. This means that for every unit increase in x, the gradient also increases by one unit. The slope of this line is constant for each point on the curve.
Suppose you are given a curve with an equation of the form y = mx + c, where m and c are constants. How would you find the gradient of this curve at any point?
Suppose you are given a curve with an equation of the form y = mx + c, where m and c are constants. How would you find the gradient of this curve at any point?
The gradient of the curve y = mx + c is always equal to the constant m, regardless of the value of x.
Describe the relationship between the gradient of a curve at a point and the slope of the tangent line at that point.
Describe the relationship between the gradient of a curve at a point and the slope of the tangent line at that point.
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If the equation of a curve is given as y = f(x), how would you find the gradient of the curve at a specific point x = a?
If the equation of a curve is given as y = f(x), how would you find the gradient of the curve at a specific point x = a?
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Study Notes
Gradient of Curves
- To find the gradient of a curve at a point, draw a tangent line at that point.
- Calculate the gradient of this tangent line.
- The gradient of the tangent line is the gradient of the curve at that point.
Finding the Gradient of a Curve
- The gradient of a curve y = x² at various x values,
- x = −3: gradient = −6
- x = −2: gradient = −4
- x = −1: gradient = −2
- x = 0: gradient = 0
- x = 1: gradient = 2
- x = 2: gradient = 4
- x = 3: gradient =6
- This suggests a relationship between x and gradient: gradient = 2x.
Relationship between x and Gradient
- A rule exists to find the gradient without drawing tangents, focusing on coordinates of a curve.
- This rule utilizes a method of calculating the gradient using a function f(x) at a point x:
- gradient = lim[h→0] (f(x + h) - f(x))/h
- Simplifying this approach leads to calculating the gradient using differentiation.
Differentiation
- Using the formula book for differentiation,
- The derivative of x² is 2x.
- The method of finding a gradient without drawing tangent lines is called differentiation.
Example 1: Point (4, 16) on y = x²
- Gradient at point (4,16):
- gradient=lim[h→0] ((4+h)² - 16)/h = 8
- The gradient at (4,16) is 8
Example 2 and 3: Differentiating other equations
- Proving that the derivative of x⁴ is 4x³.
- Proving that the derivative of 2x² - x is 4x – 1.
Retention Homework: Probability
- Probability of John buying fruit: 0.7 and probability buying vegetables: 0.4
- Probability of buying fruit or vegetables or both: 0.82 (assuming independence)
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Description
This quiz explores the concept of finding the gradient of curves using tangent lines and differentiation. It provides examples, calculations, and relationships between x-values and gradients specifically for the function y = x². Test your understanding of these fundamental calculus principles!
