Finding-Gradients-Of-Tangents Aqa Higher

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Exam code:8300

Finding gradients of tangents

The gradient of a graph at a point is equal to the gradient of the tangent to the curve at that point
- A tangent is a line that touches a curve, but does not cross it

A quadratic with two tangents drawn on it. The gradient of the curve at the point x=1 will be equal to the gradient of the purple tangent. The gradient of the curve at th epoint x=6 will be equal to the gradient of the green tangent.

How do I estimate the gradient of a curve using a tangent?

To find an estimate for the gradient of a curve at a point:
- Draw a tangent to the curve at the point
- Find the gradient of the tangent using
  - Gradient = rise ÷ run
  - or difference in y ÷ difference in x
  - In the example below, the gradient of the tangent at x = 4 would be $fraction numerator negative 2.5 over denominator 4 end fraction equals negative 0.625$
  - Remember that the rise is negative if it is going down
  - This means the gradient of the curve at x = 4 is also -0.625
It is an estimate because the tangent has been drawn by eye and is not exact
- To find the exact gradient we would need to use differentiation

What does the gradient represent?

The gradient represents the rate of change of y with x
- I.e. For every increase in x by 1, how much does y increase?
Consider the quantities used for the axes to determine the meaning of the gradient
- In a distance-time graph, the gradient is the rate of change of distance with time
  - This is the speed
- In a speed-time graph, the gradient is the rate of change of speed with time
  - This is the acceleration
- In a graph of volume against radius, e.g. as a balloon is inflated, the gradient is the rate of change of volume as the radius increases

Examiner Tips and Tricks

When drawing a tangent by hand:
- Use a ruler
- Draw the line as long as you can
When finding the gradient of the tangent:
- Pick two points that are far away from one another
- This will reduce the effect of any inaccuracy

Worked Example

The graph below shows $y equals cube root of x$ for $0 less or equal than x less or equal than 1$ .

Find an estimate of the gradient of the curve at the point where <img alt=”x equals 0.5″ data-mathml=”<math ><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true,”toolbar”:”<toolbar ref=’general’><tab ref=’general’><removeItem ref=’setColor’/><removeItem ref=’bold’/><removeItem ref=’italic’/><removeItem ref=’autoItalic’/><removeItem ref=’setUnicode’/><removeItem ref=’mtext’ /><removeItem ref=’rtl’/><removeItem ref=’forceLigature’/><removeItem ref=’setFontFamily’ /><removeItem ref=’setFontSize’/></tab></toolbar>”}</annotation></semantics></math>” height=”22″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2222%22%20width%3D%2251%22%20wrs%3Abaseline%3D%2216%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E.%3C%2Fmo%3E%3Cmn%3E5%3C%2Fmn%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math11824c643d1feb4da18b28ed527’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADxjdnQgDVUNBwAAAVgAAAA6Z2x5ZoPi2VsAAAGUAAAA%2BGhlYWQQC2qxAAACjAAAADZoaGVhCGsXSAAAAsQAAAAkaG10eE2rRkcAAALoAAAADGxvY2EAHTwYAAAC9AAAABBtYXhwBT0FPgAAAwQAAAAgbmFtZaBxlY4AAAMkAAABn3Bvc3QB9wD6AAAExAAAACBwcmVwa1uragAABOQAAAAUAAADSwGQAAUAAAQABAAAAAAABAA

Maths Gcse Aqa Higher

Finding-Gradients-Of-Tangents Aqa Higher