Problem 1 :
Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
(i) f(x) = |(1/x)|, x ∊ [-1, 1] Solution
(ii) f (x) = tan x, x ∊ [0, π] Solution
(iii) f(x) = x - 2 log x, x ∊ [2, 7] Solution
Problem 2 :
Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions :
(i) f(x) = x2 − x, x ∈ [0, 1] Solution
(ii) f(x) = (x2-2x)/(x+2), x ∈ [-1, 6] Solution
(iii) f(x) = √x - (x/3), x∈ [0, 9] Solution
Problem 3 :
Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals :
(i) f(x) = (x+1)/x, x ∊ [-1, 2] Solution
(ii) f(x) = |3x+1|, x ∊ [-1, 3] Solution
Problem 4 :
Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:
(i) f (x) = x3 - 3x + 2, x ∊ [-2, 2] Solution
(ii) f(x) = (x-2)(x-7), x ∊ [3, 11] Solution
Problem 5 :
Show that the value in the conclusion of the mean value theorem for
(i) f (x) = 1/x on a closed interval of positive numbers [a, b] is √ab Solution
(ii) f (x) = Ax2 + Bx + C on any interval [a, b] is (a + b)/2
Problem 6 :
A race car driver is racing at 20th km. If his speed never exceeds 150 km/hr, what is the maximum distance he can cover in the next two hours. Solution
Problem 7 :
Suppose that for a function f (x), f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4) − f(1) ≤ 3. Solution
Problem 8 :
Does there exist a differentiable function f(x) such that f(0) = −1, f(2) = 4 and f'(x) ≤ 2 for all x . Justify your answer. Solution
Problem 9 :
Show that there lies a point on the curve
f(x) = x(x+3)eπ/2, -3 ≤ x ≤ 0,
where the tangent is parallel to the x-axis.
Problem 10 :
Using mean value theorem prove that for a > 0, b > 0,
|e-a-e-b| < |a-b|
(1) (i) f(0) = ∞ (ii) f(π/2) = ∞ (ii) f(2) ≠ f(7)
(2) (i) c = 1/2 (ii) x = -2±2√2 (iii) x = 9/4
(3) (i) Not continuous on [-1, 2].
(ii) not differentiable at x = -1/3
(4) (i) x = ±2/√3 (ii) x = 7
(5) (i) x = √ab (ii) x = (a+b)/2
(6) 320 km/hr.
(7) [f(4)-f(1)] ≤ 3
(8) f'(x) = 2.5, it is not possible, because it is not in the interval [0, 2].
(9) x = -3/2,
(10) |[e-b - e-a]| ≤ |b-a|
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