APPLICATION: (I)TANGENT LINE (II)RELATED RATES (III)MINIMUM AND MAXIMUM VALUES CHAPTER 4

Tangent line Consider a function, with point lying on the graph:  Tangent line to the function at is the straight line that touches at that point.  Normal line is the line that is perpendicular to the tangent line. Tangent Line Normal Line

Tangent Line Equation: or Normal Line Equation: or

Example 1: 1.Find the slope of the curve at the given points 2.Find the lines that are tangent and normal to the curve at the given point.

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. RELATED RATES

Example 2 Suppose that the radius, r and area, of a circle are differentiable functions of t. Write an equation that relates to. Answer:

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: Differentiate (1) wrt t: 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: Differentiate wrt t : r

Exercise 1 A 13ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5ft/s. (a)How fast is the top of the ladder sliding down the wall? (b)At what rate is the area of the triangle formed by the ladder, wall and ground changing (c)At what rate is the angle between the ladder and the ground changing?

Exercise 2 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 (a)The area (b)The perimeter

Exercise 3: 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?

Use 1 st 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 MAXIMUM & MINIMUM

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

Let f be a function defined on an interval I and let x 1 and x 2 be any two points in I 1)If f (x 1 )< f (x 2 ) whenever x 1 < x 2, then f is said to be increasing on I 2)If f (x 1 )> f (x 2 ) whenever x 1 < x 2, then f is said to be decreasing on I INCREASING & DECREASING

Suppose that f is continuous on [a,b] and differentiable on (a,b). 1)If f’(x)>0 at each point, then f is said to be increasing on [a,b] 2)If f’(x)<0 at each point, then f is said to be decreasing on [a,b] 1 st DERIVATIVE TEST

The graph of a differentiable function y=f(x) 1)Concave up on an open interval if f’ is increasing on I 2)Concave down on an open interval if f’ is decreasing on I CONCAVITY

Let y=f(x) be twice-differentiable on an interval I 1)If f”(x)>0 on I, the graph of f over I is concave up 2)If f”(x)<0 on I, the graph of f over I is concave down 2 ND DERIVATIVE TEST: TEST FOR CONCAVITY 2 ND DERIVATIVE TEST: TEST FOR CONCAVITY

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

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

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:

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

Step 3: 1 st and 2 nd Derivative Test Interval Test Value, x IncreasingDecreasing Increasing --++ ConcaveConcave down Concave up

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

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