List of top Mathematics Questions on Application of derivatives asked in BITSAT

If the function \( f(x) \), defined below, is continuous on the interval \([0,8]\), then: \[ f(x) = \begin{cases} x^2 + ax + b, & 0 \leq x < 2 \\ 3x + 2, & 2 \leq x \leq 4 \\ 2ax + 5b, & 4 < x \leq 8 \end{cases} \]

If \( \frac{dy}{dx} - y \log_e 2 = 2^{\sin x} (\cos x - 1) \log_e 2 \), then \( y \) is:

The solution of the differential equation \( (x + 1)\frac{dy}{dx} - y = e^{3x}(x + 1)^2 \) is:

If the area bounded by the curves \( y = ax^2 \) and \( x = ay^2 \) (where \( a>0 \)) is 3 sq. units, then the value of \( a \) is:

The area of the region bounded by the curves \( x = y^2 - 2 \) and \( x = y \) is:

The line \(y = mx\) bisects the area enclosed by lines \(x = 0\), \(y = 0\), and \(x = \frac{3}{2}\) and the curve \(y = 1 + 4x - x^2\). Then, the value of \(m\) is:

The population \( p(t) \) at time \( t \) of a certain mouse species satisfies the differential equation:
\[ \frac{d p(t)}{dt} = 0.5p(t) - 450. \] If \( p(0) = 850 \), then the time at which the population becomes zero is:

The point of inflexion for the curve \(y = (x - a)^n\), where \(n\) is odd integer and \(n \ge 3\), is:

The altitude of a cone is 20 cm and its semi-vertical angle is \(30^\circ\). If the semi-vertical angle is increasing at the rate of \(2^\circ\) per second, then the radius of the base is increasing at the rate of:

If the angle made by the tangent at the point \((x_0, y_0)\) on the curve \(x = 12(t + \sin t \cos t)\), \(y = 12(1 + \sin t)^2\), with \(0<t<\frac{\pi}{2}\), with the positive x-axis is \(\frac{\pi}{3}\), then \(y_0\) is equal to:

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is:

At \( x = \frac{\pi^2}{4} \), \( \frac{d}{dx} \left( \tan^{-1}(\cos\sqrt{x}) + \sec^{-1}(e^x) \right) = \)

If \( y = \tan^{-1}\left( \frac{\sqrt{x} - x}{1 + x^{3/2}} \right) \), then \( y'(1) \) is equal to:

If \( x\sqrt{1 + y} + y\sqrt{1 + x} = 0 \), then find \( \frac{dy}{dx} \).

Match List I with List II and select the correct answer using the code given below the lists. List I • [(A)] f(x)=cos x • [(B)] f(x)=ln x • [(C)] f(x)=x²-5x+4.3 • [(D)] f(x)=eˣ List II • [1.] The graph cuts y-axis in infinite number of points • [2.] The graph cuts x-axis in two points • [3.] The graph cuts y-axis in only one point • [4.] The graph cuts x-axis in only one point • [5.] The graph cuts x-axis in infinite number of points

For any differentiable function y of x, (d²x)/(dy²)((dy)/(dx))³+(d²y)/(dx²)=

If y=x+√(1+x²), then (1+x²)dfracd²ydx²+x(dy)/(dx) is

If
\[ f(x) = \cos^{-1} \left[ \frac{1 - (\log x)^2}{1 + (\log x)^2} \right], \]
then the value of \(f'(e)\) is equal to

The number of real roots of the equation \[ e^{x-1}+x-2=0 \] is

If \(g\) is the inverse of function \(f\) and \(f'(x)=\sin x\), then \(g'(x)\) is equal to

For the function \[ f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} - x^2 + x + 1, \] \( f'(1) = mf'(0) \), where \( m \) is equal to: