Chapter 6: Application of Derivatives
Q1. Find the slope of the tangent to the curve y = x³ − x + 1 at the point whose x-coordinate is 2.
Solution
dy/dx = 3x² − 1
At x = 2: dy/dx = 3(4) − 1 = 11
Q2. Find the equation of the tangent to the curve y = x² − 2x + 3 which is parallel to the line 2x − y + 9 = 0.
Solution
Slope of given line = 2. For tangent to be parallel, dy/dx = 2.
dy/dx = 2x − 2 = 2 ⇒ 2x = 4 ⇒ x = 2
At x = 2: y = 4 − 4 + 3 = 3. Point is (2, 3).
Equation: y − 3 = 2(x − 2) ⇒ 2x − y − 1 = 0
Q3. Find the intervals in which the function f(x) = 2x³ − 9x² + 12x − 5 is strictly increasing.
Solution
f'(x) = 6x² − 18x + 12 = 6(x² − 3x + 2) = 6(x−1)(x−2)
f'(x) > 0 when (x−1)(x−2) > 0
∴ f is strictly increasing on (−∞, 1) ∪ (2, ∞)
Q4. Find the approximate value of (15)1/4 using differentials.
Solution
Let f(x) = x1/4, x = 16, Δx = −1
f'(x) = (1/4)x−3/4
f(16) = 161/4 = 2
f'(16) = (1/4)(16)−3/4 = (1/4)(1/8) = 1/32
f(15) ≈ f(16) + f'(16)(−1) = 2 − 1/32 = 63/32 ≈ 1.96875
Q5. Find the local maxima and minima of f(x) = x³ − 3x + 2.
Solution
f'(x) = 3x² − 3 = 3(x² − 1) = 3(x−1)(x+1)
f'(x) = 0 ⇒ x = 1 or x = −1
f''(x) = 6x
f''(1) = 6 > 0 ⇒ Local minimum at x = 1, f(1) = 0
f''(−1) = −6 < 0 ⇒ Local maximum at x = −1, f(−1) = 4
Q6. A stone is dropped into a quiet lake and ripples move in a circle. Find the rate of change of the area with respect to the radius when the radius is 5 cm.
Solution
A = πr²
dA/dr = 2πr
At r = 5: dA/dr = 2π(5) = 10π cm²/cm