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# Calculus 11.2

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### Calculus 11.2

1. 1. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points If I have seen further….. …. it is because I have stood on the shoulders of giants. ISaac Newton to Robert Hooke in 1675 ISAAC NEWTON 1643 -1727
2. 2. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
3. 3. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points 1
4. 4. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
5. 5. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction The two-point formula Gradients of secants The concept of gradient is so important for a thorough understanding of differential calculus. The graphs of some linear functions are steep with a positive slope > The graphs of some linear functions are less steep > … and others have negative slopes > Gradient is a measure of this steepness or slope. It is defined as the ratio of the rise to the run. The gradient of the green function is 2. Check the gradient of the red function is 1/3 And the blue line has a gradient of-1 2
6. 6. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction The two-point formula Gradients of secants The given line passes through the points P and Q where: P = ( 2, 3 ) P Q = ( 1, 1 ) X In the interval PQ: rise rise = 3 - 1 run = 2 - 1 Q X run Generally, the straight line passing through the two points; , Has a gradient given by: 3
7. 7. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction The two-point formula Gradients of secants Using the formula for the gradient of a lineHere is the graph of the function through two points, we have: And here is a secant PQ where and . 4
8. 8. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
9. 9. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Case study An algebraic approach Limiting process The gradient of the graph of a linear function is easy to find; we can use the two- point formula as shown in the previous section. But how can we find the gradient at different points on a non-linear function, such as the one shown here? Clearly the parabola gets steeper as the x-values increase… … but how can we measure the actual gradient at any particular point on the curve? Gradient of the function at the point ( 3, 9) > The gradient at a point P on the DEFINITION curve is defined as the gradient of the tangent to the curve at Gradient of the function at the point ( 2, 4) that point. 5
10. 10. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Case study An algebraic approach Limiting process Our goal here is means finding the gradient of the tangent By definition thisto find exactly the gradient of the function to the curve the point at at that point… As a first approximation, As a second approximation, For a third approximation consider the secant AP consider the secant BP we will need to zoom in and consider secant CP…. Note now how close the tangent is to the curve C P 6
11. 11. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Case study An algebraic approach Limiting process Our goal here is to find exactly the gradient of the function at the point By definition this means finding the gradient of the tangent to the curve at that point… As a first approximation, consider the secant h x+h 7
12. 12. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Case study An algebraic approach Limiting process Using first principles find the derived function for h x+h 8
13. 13. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Case study An algebraic approach Limiting process Using first principles find the derived function for Using first principles find the derived function for SUMMARY 8
14. 14. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
15. 15. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Equation of a tangent Equation of a normal ACTIVITY 1A tangent to a curve is a straight line touching the curve at a single point. A normal is a straight line, perpendicular to the tangent. FIGURE 1 FIGURE 2 FACT SHEET This diagram shows the TANGENT to the This diagram shows the NORMAL to the • A TANGENT touches a curve at a single curve curve point at the point (1, -2) at the point (1, -2) • It’s gradient, , is given by the gradient or derived function at the value • Its equation is given by NORMAL • The NORMAL at a point is perpendicular to the tangent at that point. (It’s at 90 degrees) TANGENT • It’s gradient, , is found using the gradient of the tangent and the fact that • It’s equation is given by 9
16. 16. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Equation of a tangent Equation of a normal ACTIVITY 1Find the equation of the tangent to the curve at the point (2, 0) METHOD 1 Differentiate to obtain the gradient function 2 Find the gradient of the function at x=2 3 Substitute the gradient, m = 1 and the coordinates of the point into the point/ gradient form of a straight line. 10
17. 17. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Equation of a tangent Equation of a normal ACTIVITY 1Find the equation of the normal to the curve at the point (2, 0) METHOD 1 Differentiate to obtain the gradient function 2 Find the gradient of the function at x=2 3 Find the gradient of the normal at x=2 4 Substitute the gradient, m = -1 and the coordinates of the point into the point/ gradient form of a straight line. 11
18. 18. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsIntroduction Equation of a tangent Equation of a normal ACTIVITY
19. 19. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
20. 20. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
21. 21. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
22. 22. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
23. 23. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals 2 Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
24. 24. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
25. 25. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
26. 26. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
27. 27. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
28. 28. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points MAXIMUM MINIMUM INFLEXION STATIONARY POINT A STATIONARY POINT B POINT C A LOCATED AT: LOCATED AT: LOCATED AT: C B FIRST DERIVATIVE IS ZERO FIRST DERIVATIVE IS ZERO SECOND DERIVATIVE IS NEGATIVE SECOND DERIVATIVE IS POSITIVE SECOND DERIVATIVE IS ZERO
29. 29. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points FIGURE 1 FIGURE 2 FACT SHEET This diagram shows the STATIONARY POINT on This diagram shows the STATIONARY POINTS on • Stationary points lie on the graphs of functions where the graph of the quadratic function the graph of the cubic function the gradient is zero. (The tangents to the curve are horizontal at these points; the function is neither increasing nor decreasing.) • The stationary point in figure 1 is called a maxima. • Figure 2 shows a function having both a maxima and minima
30. 30. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points Find the stationary point on the graph of the function METHOD 1 Find the gradient function by differentiating 2 Find the x-value for which the gradient is zero by solving the equation 3 Find the y-value of the function at x = 1.5 by substitution into the original function 4 Write down the coordinates of the stationary point
31. 31. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary pointsFind the stationary points on the function and determine their nature METHOD 1 Differentiate the given function to obtain the gradient function 2 Find the x-values for which the gradient function is zero. 3 Substitute these x-values into the original function to determine the stationary points. 4 Find the sign of the second derivative to determine the nature of each stationary point.
32. 32. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points
33. 33. DIFFERENTIAL CALCULUSHome Limits Gradient First principles Tangents and Normals Graph sketching Derivatives of functions Composite functions Second derivatives Stationary points