WEEK THREE SS2 FURTHER MATHS SECOND TERM DIFFERENTIATION 3

WEEK THREE

SS2 FURTHER MATHS SECOND TERM

DIFFERENTIATION 3

CONTENT

(a) Application of Differentiation

(b) Differentiation of Implicit Functions.

SUB TOPIC: APPLICATION OF DIFFERENTIATION

Examples:

1. Suppose an object is dropped from rest from a given height, the distance s through which the object has dropped after t seconds (ignoring air resistance) is given by the formula

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1. Let p(t) denote the population of a colony of bacteria after t hours. If how fast is the population growing after 3hours?

Solution:

1. We want the instantaneous rate of growth of growth of the population when t = 3hours.

CLASS ACTIVITY:

1. The distance (d metres) fallen by a stone in t seconds is given by the equation
2. Find the maximum and minimum points of the curve and sketch the curve.

SUB TOPIC: IMPLICIT DIFFERENTIATION

Functions which have their derivatives in terms of it may be possible to separate completely from to different side. Such functions are called implicit function.

Examples are ²²

Example:

1. If ³²³³ find

Solution:

²

²²

Solving for

) ²

.

1. If ⁴³⁴² find

Solution:

Differentiating both side with respect to

³³³⁴²)

³⁴²)

Solving for

³³

CLASS ACTIVITY:

Differentiate the following:

1. ²²³³
2. ³²+
3. ²

PRACTICE EXERCISE:

Essay:

1. If
2. Given that
3. ²
4. A ball rolls down an inclined plane. It travels s centimeters in t seconds and (i) how far has the ball travelled in the first second? (ii) what is its speed at the end of the fourth second?
5. A particle moves along a straight so that after a time t seconds, its distance Scm from the starting point O is given by find: (a) the distances from O when the particle is momentarily at rest; (b) the velocity when the acceleration is zero.

ASSIGNMENT

1. Differentiate with respect to x:
2. Find from first principle, the derivative, with respect to x of the function
3. Find
4. Find
5. If calculate the average rate of change(gradient) of y, w.r.t x, in the interval between x = c and x = c + h. hence find . Also find the maximum and minimum points of the curve defined by and sketch the curve.

KEY WORDS

• LIMIT
• CONTINUITY
• FUNCTION
• DERIVATIVE
• DIFFERENTIATION
• FIRST PRINCIPLE
• WITH RESPECT TO
• POLYNOMIALS
• GRADIENT
• TRANSCENDAL
• RATE OF CHANGE
• HIGHER DERIVATIVE