Note 1:
We cannot obtain c if any one of the conditions of Rolle's theorem are not satisfie..
Example:
Verify Rolle's theorem for the function f (x) = x 2 - 8x + 12 on (2, 6) Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied. f (2) = 2 2 - 8 (2) + 12 = 0 f (6) = 36 - 48 + 12 = 0 ..
Verify Rolle's theorem for the function f (x) = x 2 - 8x + 12 on (2, 6) Since a polynomial function is continuous and differentiable everywhere f (x) is differentiable and continuous (i) and (ii) conditions of Rolle's theorem is satisfied. f (2) = 2 2 - 8 (2) + 12 = 0 f (6) = 36 - 48 + 12 = 0 ..Step 1:
Show that the function is continuous in the given interval. Some known standard functions which are continuous, can be mentioned directl..
Step 3:
Check if f (a) = f (b) If all the above condition are satisfied, then Rolle's theorem is applicable else the Rolle's theorem is not applicable. If Rolle's theorem is applicable, solve f '(c) = 0. Show that one of these roots lie in the open interval (a, b..
Note 1:
If m = 0 the tangent is parallel to x-axis. ..
If m = 0 the tangent is parallel to x-axis. ..Note 4:
Equation of the normal at (x 1 ,y 1 ) is o..
Equation of the normal at (x 1 ,y 1 ) is o..Solution:
Since the tangent to the given curve is parallel to the line 4x - 2y + 5 = 0 Slope of the tangent = Slope of the given line On simplification, the equation is 48 x - 24y = 2..
Since the tangent to the given curve is parallel to the line 4x - 2y + 5 = 0 Slope of the tangent = Slope of the given line On simplification, the equation is 48 x - 24y = 2..Applications of Definite Integrals
Let y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given byLet y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given by (ii) The area bounded by the curve x = f (y) y = axis and the abscissae at..
Let y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given byLet y = f (x) be a curve. The area bounded by y = f (x), x-axis and the ordinates at x = a and x = b is given by (ii) The area bounded by the curve x = f (y) y = axis and the abscissae at..Note 2:
If curve f (x) lies above the x-axis and g (x) lies below the x-axis then area bounded by f (x) and g (x) is ..
If curve f (x) lies above the x-axis and g (x) lies below the x-axis then area bounded by f (x) and g (x) is ..Example:
Find the area of the region between the X-axis and the graph of f(x) = x 3 - x 2 - 2x -1 x < ..
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