Application of Derivatives - I

We can use derivatives to measure rate of change of quantities, to find increasing and decreasing functions, to find tangents and normals etc.
•    If a quantity y varies with another quantity x, satisfying some rule y = f(x), then dy/dx represents the rate of change of y with respect to x.

•    If two variables x and y are varying with respect to another variable t, i.e., if y = f(t) and x = g(t), then dy/dx = (dy/dt)/(dx/dt), provided that dx/dt is not equal to 0.

•    A function f is said to be increasing function on an interval (a, b) if x1 < x2 in (a, b)
 f(x1) ≤ f(x2) for all x1, x2∈ (a, b).

•    A function f is said to be decreasing function on an interval (a, b) if x1 < x2 in (a, b)
 f(x1) ≥ f(x2) for all x1, x2∈ (a, b).

•    If f(x) be a function continuous on [a, b] and differentiable on (a, b), then
f(x) is strictly increasing  in (a, b) if f´(x) > 0 for all x in (a, b).  
 f(x) is strictly decreasing in  (a, b) if f´(x) < 0 for all x in (a, b).

•    The equation of tangent at (x1, y1) to a curve y = f(x) is given by
       y - y1 =  (x - x1) f' (x1).

•    The equation of normal at (x1, y1) to a curve y = f(x) is given by  
      y - y1 =  - (x - x1)/ f' (x1).
 

Keywords: Derivative, Rate of Change, increasing functions, decreasing functions, tangents and normals

 

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  • Q1

    Marks:1
    Answer:

    (e, ).

    Explanation:

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  • Q2

    The function f(x) = cos x is

    Marks:1
    Answer:

    strictly decreasing on (0, ).

    Explanation:
    Given f(x) = cos x  f'(x) = -sin x < 0 x  (0, ) (Becasue sin x > 0 x  (0, ))

     

     

     f'(x) < 0 x  (0, )

     

     

      f(x) is strictly decreasing on (0, ).

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  • Q3

    On R, the function f(x) = 7x – 3 is

    Marks:1
    Answer:

    strictly increasing.

    Explanation:

    Given f(x) = 7x - 3, x  R  f'(x) = 7 > 0, x   R

     f is strictly increasing on R.

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  • Q4

    The function f(x) = 2 – 3x is

    Marks:1
    Answer:

    strictly decreasing.

    Explanation:
    Given f(x) = 2 – 3x, x  R.

    f’(x) = – 3 < 0 

     f is strictly decreasing on R.

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  • Q5

    The function  f(x)= x2+2x+3 increases in the interval

    Marks:1
    Answer:

    (–1, )

    Explanation:

    We have f(x)= x2+2x+3

    f'(x)= 2x+2 

    For turning point f'(x)=0 x= -1

    f'(x)> 0 if x  (–1, )

    f (x) is increasing in the interval (–1, ).

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