If y = x sinx, find dy/dx when x = π/2.
-π/2
-1
1
π/2
Correct answer is C
y = x sin x
dy/dx = 1 x sinx + x cosx
= sinx + x cosx
At x = π/2, = sin (π/2) + (π/2) cos (π/2)
= 1 + (π/2) x (0) = 1
Find the rate of change of the volume, V of a sphere with respect to its radius, r when r = 1
12π
4π
24π
8π
Correct answer is B
Volume of sphere, V = 4/3 x πr3
Rate of change of V = dv/dr
Thus if V = 4/3 x πr3,
=> dv/dr = 4πr2
At r = 1, Rate = 4 x π x 1 = 4π
Find the area bounded by the curves y = 4 - x2 and y = 2x + 1
20(1/3) sq. units
20(2/3) sq. units
10(2/3) sq. units
10(1/3) sq. units
Correct answer is C
y = 4 - x\(^2\) and y = 2x + 1
=> 4 - x2 = 2x + 1
=> x2 + 2x - 3 = 0
(x+3)(x-1) = 0
thus x = 1 or x = -3.
Integrating x\(^2\) + 2x - 3
Differentiate \((2x+5)^{2} (x-4)\) with respect to x.
4(2x+5)(x-4)
4(2x+5)(4x-3)
(2x+5)(2x-13)
(2x+5)(6x-11)
Correct answer is D
\(y = (2x + 5)^{2} (x - 4)\)
\(\frac{\mathrm d y}{\mathrm d x} = (2x + 5)^{2} (1) + (x - 4)(2)(2)(2x + 5)\)
= \((2x + 5)(2x + 5 + 4x - 16)\)
= \((2x + 5)(6x - 11)\)
Evaluate \(\int 2(2x - 3)^{\frac{2}{3}} \mathrm d x\)
3/5(2x-3)5/3 + k
6/5(2x-3)5/3 + k
2x-3+k
2(2x-3)+k
Correct answer is A
\(\int 2(2x - 3)^{\frac{2}{3}} \mathrm d x\)
Let \(u = 2x - 3\)
\(\mathrm d u = 2 \mathrm d x\)
= \(\int u^{\frac{2}{3}} \mathrm d u\)
= \(\frac{u^{\frac{5}{3}}}{\frac{5}{3}} + k\)
= \(\frac{3}{5} (2x - 3)^{\frac{5}{3}} + k\)