The point on the curve 6y = x³ + 2 at which y-coordinate changes 8 times as fast as x-coordinate is 

The abscissa of the point on the curve \(f\left( x \right) =\sqrt { 8-2x } \) at which the slope of the tangent is -0.25 ?

The slope of the line normal to the curve f(x) = 2cos 4x at \(x=\cfrac { \pi }{ 12 } \) is

The point on the curve y = x² is the tangent parallel to X-axis is __________

The equation of the tangent to the curve y = x²-4x+2 at (4, 2) is __________

A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.
(i) At what times the particle changes direction?
(ii) Find the total distance travelled by the particle in the first 4 seconds.
(iii) Find the particle’s acceleration each time the velocity is zero.

Find the local extrema for the following function using second derivative test:
f(x) = x² e^-2x

Find the maximum and minimum values of f(x) = |x+3| ∀ \(x\in R\).

Find the intervals of increasing and decreasing function for f(x) = x³ + 2x² - 1.

A particle is fired straight up from the ground to reach a height of s feet in t seconds, where s(t) = 128t −16t².
(1) Compute the maximum height of the particle reached.
(2) What is the velocity when the particle hits the ground?

If we blow air into a balloon of spherical shape at a rate of 1000 cm³ per second. At what rate the radius of the baloon changes when the radius is 7cm? Also compute the rate at which the surface area changes.

Prove using the Rolle’s theorem that between any two distinct real zeros of the polynomial \(a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\) there is a zero of the polynomial \(na_{n}x^{n-1}+(n-1)a_{n-1}x^{n-2}+...+a_{1}\)

Find the equation of normal to the cure y = sin²x at \(\left( \frac { \pi }{ 3 } ,\frac { 3 }{ 4 } \right) \).

Verify LMV theorem for f(x) = x³ - 2x² - x + 3 in [0, 1].

Find the equation of the tangent and normal to the Lissajous curve given by x = 2cos 3t and y = 3sin 2t, t ∈ R

If the curves ax²+ by² = 1 and cx²+ dy² = 1 intersect each other orthogonally then, \(\frac{1}{a}-\frac{1}{b}=\frac{1}{c}-\frac{1}{d}\)