APPLICATION: RELATED RATES MINIMUM AND MAXIMUM VALUES CHAPTER 4 APPLICATION: RELATED RATES MINIMUM AND MAXIMUM VALUES

RELATED RATES A process of finding a rate at which a quantity changes by relating that quantity to the other quantities. The rate is usually with respect to time, t.

Example 3 How fast is the area of a rectangle changing from one side 10cm long and the side increase at a rate of 2cm/s and the other side is 8cm long and decrease at a rate of 3cm/s?

Solution: x y

Example 4 A stone is dropped into a pond, the ripples forming concentric circles which expand. At what rate is the area of one of these circles increasing when the radius is 3m and increasing at the rate of 0.6ms-1?

Solution: r

Exercise 1 The length l of a rectangle is decreasing at the rate of 2cm/s, while the width w is increasing at the rate 2cm/s. When l=12cm and w=5cm find the rates of change The area The perimeter

Exercise 2: When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01cm/min. At what rate is the plate’s area increasing when the radius is 50cm?

MAXIMUM & MINIMUM Use 1st derivative to locate and identify extreme values(stationary values) of a continuous function from its derivative Definition: Absolute Maximum and Absolute Minimum Let f be a function with domain D. Then f has an ABSOLUTE MAXIMUM value on D at a point c if: ABSOLUTE MINIMUM

STATIONARY POINT A point on the graph of a function y = f(x) where the rate of change is zero. Example 6 Find stationary points:

INCREASING & DECREASING Let f be a function defined on an interval I and let x1 and x2 be any two points in I If f (x1)< f (x2) whenever x1 < x2, then f is said to be increasing on I If f (x1)> f (x2) whenever x1 < x2, then f is said to be decreasing on I

1st DERIVATIVE TEST Suppose that f is continuous on [a,b] and differentiable on (a,b). If f’(x)>0 at each point , then f is said to be increasing on [a,b] If f’(x)<0 at each point , then f is said to be decreasing on [a,b]

CONCAVITY The graph of a differentiable function y=f(x) Concave up on an open interval if f’ is increasing on I Concave down on an open interval if f’ is decreasing on I

2ND DERIVATIVE TEST: TEST FOR CONCAVITY Let y=f(x) be twice-differentiable on an interval I If f”(x)>0 on I, the graph of f over I is concave up If f”(x)<0 on I, the graph of f over I is concave down

MAXIMUM POINT & MINIMUM POINT If y is minimum Therefore (x,y) is a minimum point. If y is maximum Therefore (x,y) is a maximum point.

CONCAVITY A point where the graph of a function has a tangent line and where the concavity changes is a POINT OF INFLEXION.

Example 5: Find y’ and y” and then sketch the graph of y=f(x)

Solution: Step 1: Find the stationary point Therefore, the stationary points are: Stationary values

The 1st Derivative Test Since the stationary values are : The intervals: Table: x Interval x = c Sign of Conclusion -3 + Increasing - Decreasing 2

Step 2 : Find inflexion point Therefore, the inflexion points is:

The 2nd Derivative Test Since the inflexion value is : The intervals: Table: x = c Sign of Conclusion -2 - Concave Downwards + Concave Upwards

Step 4: Test for maximum and minimum At Therefore, (-2, 17) is a maximum point. Therefore, (4/3, -41/27) is a minimum point.

Example 6: Find y’ and y” and then sketch the graph of y=f(x)