4
,5
,6
, APPLICATION OF DERIVATIV
1. Common application of derivatives.
• Finding Rate of Change of a Quantity • Finding Maxima and Minima, a
• Finding the Approximation Value • Determining Increasing and D
• Finding the equation of a Tangent and Normal To a Curve
(2) Approximations (3) Equations of ta
Assume we have a function y = f(x), which is defined
• Equation of tangent at (x1, y1)
in the interval [a, a + h], then the average rate of change in the
function in the given interval is (f(a + h) – f(a))/h • Equation of Normal at (x1, y1)
Now using the definition of derivative, we can write
y – y1= − dy ( x − x1 )
f (a + h ) − f (a ) dx ( x1 , y1 )
f ′ ( a ) = lim
h →0 h
which is also the instantaneous rate of change of the function
f(x) at a. Now, for a very small value of h, we can write
f ′( a ) ≈ (f ( a + h ) − f (a )) / h
(4) Increasing decreasing
functions.
(5) Maxima
Properties of monotonic functions :
(1) If f(x) is continuous on [a, b] such that f' (c) 0 for each c,
then f(x) is monotonically increasing function. Similar definition
goes for monotonically decreasing function. Poin
(2) If f(x) is strictly increasing function on [a, b] then f–1(x) exists Poin
& is also strictly increasing on [a, b]. Similar result follows for
strictly decreasing functions.
(3) If f(x) & g(x) are two continuous & differentiable functions,
then we can relate fog (x) & gof (x) by the following table Global maxima, glob
f (x) g(x) Maximum = max {f(a)
+ denotes increasing function
fog/gof
+ + +
– denotes decreasing function + − − Minimum = min {f(a),
− + −
− − + c1 c2…. cn are n critical
(6) Test Of Local Maxima & Mi
,5
,6
, APPLICATION OF DERIVATIV
1. Common application of derivatives.
• Finding Rate of Change of a Quantity • Finding Maxima and Minima, a
• Finding the Approximation Value • Determining Increasing and D
• Finding the equation of a Tangent and Normal To a Curve
(2) Approximations (3) Equations of ta
Assume we have a function y = f(x), which is defined
• Equation of tangent at (x1, y1)
in the interval [a, a + h], then the average rate of change in the
function in the given interval is (f(a + h) – f(a))/h • Equation of Normal at (x1, y1)
Now using the definition of derivative, we can write
y – y1= − dy ( x − x1 )
f (a + h ) − f (a ) dx ( x1 , y1 )
f ′ ( a ) = lim
h →0 h
which is also the instantaneous rate of change of the function
f(x) at a. Now, for a very small value of h, we can write
f ′( a ) ≈ (f ( a + h ) − f (a )) / h
(4) Increasing decreasing
functions.
(5) Maxima
Properties of monotonic functions :
(1) If f(x) is continuous on [a, b] such that f' (c) 0 for each c,
then f(x) is monotonically increasing function. Similar definition
goes for monotonically decreasing function. Poin
(2) If f(x) is strictly increasing function on [a, b] then f–1(x) exists Poin
& is also strictly increasing on [a, b]. Similar result follows for
strictly decreasing functions.
(3) If f(x) & g(x) are two continuous & differentiable functions,
then we can relate fog (x) & gof (x) by the following table Global maxima, glob
f (x) g(x) Maximum = max {f(a)
+ denotes increasing function
fog/gof
+ + +
– denotes decreasing function + − − Minimum = min {f(a),
− + −
− − + c1 c2…. cn are n critical
(6) Test Of Local Maxima & Mi
