A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t². The stone reaches the maximum height in time t seconds is given by

The tangent to the curve y² - xy + 9 = 0 is vertical when 

What is the value of the limit \(\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right) \text { is }\) 

The function sin⁴ x + cos⁴ x is increasing in the interval

The number given by the Mean value theorem for the function \(\frac { 1 }{ x } \), x ∈ [1, 9] is

One of the closest points on the curve x² - y² = 4 to the point (6, 0) is

The point of inflection of the curve y = (x - 1)³ is

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

Equation of the normal to the curve y = 2x²+3 sin x at x = 0 is __________

The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are __________

The equation of the tangent to the curve x = t cost, y = t sin t at the origin is __________

In LMV theorem, we have f'(x₁) = \(\frac { f(b)-f(a) }{ b-a } \) then a < x₁ _________

If the curves y = 2e^x and y = ae^-x intersect orthogonally, then a = _________

\(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

The statement "If f has a local extremum at c and if f'(c) exists then f'(c) = 0" is ________

A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) = 100 × (1− 0.1t)², 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units.

Find the slope of the tangent to the following curves at the respective given points
y = x⁴ + 2x² − x at x = 1

Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7.

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals.
\(f(x)=x-2logx, x\in [2,7]\)

Using the Rolle’s theorem, determine the values of x at which the tangent is parallel to the x -axis for the following functions:
f(x) = x² − x, x ∈ [0, 1]

At what point on the curve y = x² on [-2, 2] is the tangent parallel to X-axis?

Find the point on the parabola y²=18x at which the ordinate increases at twice the rate of the abscissa.

Using Rolle’s theorem find the value of c for f(x) = sin x in[0,2π]

Evaluate the following limits, if necessary using L’Hopitalrule
(i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
(ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
(iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
(iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

A point moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.
(i) Find the average velocity of the points between t = 3 and t = 6 seconds.
(ii) Find the instantaneous velocities at t = 3 and t = 6 seconds.

A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres.

A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?

Find the points on the curve y² - 4xy = x² + 5 for which the tangent is horizontal.

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 absolute extrema of the following functions on the given closed interval.
\(f(x)=2cosx+sin2x;\left[ 0,\frac { \pi }{ 2 } \right] \)

Suppose that for a function f(x), f'(x) ≤ 1for all 1 ≤ x ≤ 4. Show that f(4) - f(1) ≤ 3.

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

A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t² where s
is in metres and t is in seconds.
(i) Find the acceleration at t = 1
(ii) Find velocity at t = 1
(iii) Find the maximum height reached by the ball?

The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate then find the ratio of the change of their areas.

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.
(i) How long does the camera fall before it hits the ground?
(ii) What is the average velocity with which the camera falls during the last 2 seconds?
(iii) What is the instantaneous velocity of the camera when it hits the ground?

A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep?

A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall.
(i) How fast is the top of the ladder moving down the wall?
(ii) At what rate, the area of the triangle formed by the ladder, wall and the floor is changing?

A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car?

Write the Maclaurin series expansion of the following function
log(1 - x); -1 ≤ x < 1

missle fired from ground level rises x metres vertically upwards in t seconds and \(x=100t-\frac { 25 }{ 2 } { t }^{ 2 }\). Find the 
(i) initial velocity of the missile
(ii) the time when the height of the missile is maximum
(iii) the maximum height reached
(iv) the velocity which the missile strikes the ground.

A manufacturer can sell x items at a price of rupees \(\left( 5-\frac { x }{ 100 } \right) \) each. The cost price of x items is Rs.\(\left( \frac { x }{ 5 } +500 \right) \) .Find the numbers of items he should sell to earn maximum profit.