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?

A particle moves along a horizontal line such that its position at any time t ≥ 0 is given by s(t) = t³ − 6t² +9 t +1, where s is measured in metres and t in seconds?
(1) At what time the particle is at rest?
(2) At what time the particle changes its direction?
(3) Find the total distance travelled by the particle in the first 2 seconds.

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.

Salt is poured from a conveyer belt at a rate of 30 cubic metre per minute forming a conical pile with a circular base whose height and diameter of base are always equal. How fast is the height of the pile increasing when the pile is 10 metre high?

A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and car B is driving east along the second road. At a particular time car A 10 kilometres to the north of P and traveling at 80 km/hr, while car B is 15 kilometres to the east of P and traveling at 100 km/hr. How fast is the distance between the two cars changing?

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 camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s =16t² in t seconds
What is the average velocity with which the camera falls during the last 2 seconds?

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.

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A particle moves along a line according to the law s(t) = 2t³ − 9t² +12t − 4, where t ≥ 0.

A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shore line. How fast is the beam moving along the shore line when it makes an angle of 45° with the shore?

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 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.

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?

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

Find the angle between y = x² and y = (x − 3)².

Find the angle between the curves y = x² and x = y² at their points of intersection (0,0) and (1,1).

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

Prove that the ellipse x² + 4y² = 8 and the hyperbola x²-2y² = 4 intersect orthogonally.

Find the points on the curve y = x³ − 6x² + x + 3 where the normal is parallel to the line x + y = 1729.

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

Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x +12y = 12.

Find the equations of the tangents to the curve y = \(\frac{x+1}{x-1}\) which are parallel to the line x + 2y = 6.

Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ R at any point on the curve.

Find the angle between the rectangular hyperbola xy = 2 and the parabola x² + 4y = 0

Show that the two curves x² − y² = r² and xy = c² where c, r are constants, cut orthogonally

Find the intervals of monotonicities and hence find the local extremum for the following function:
f(x) = 2x³+ 3x²-12x

Find the intervals of monotonicities and hence find the local extremum for the following function:
\(f(x)=\frac { x }{ x-5 } \)

Find the intervals of monotonicities and hence find the local extremum for the following function:
\(f(x)=\frac { { e }^{ x } }{ 1-{ e }^{ x } } \)

Find the intervals of monotonicities and hence find the local extremum for the following function:
\(f(x)=\frac { { x }^{ 3 } }{ 3 } -logx\)

Find the intervals of monotonicities and hence find the local extremum for the following function:
f(x) = sin x cos x + 5, x ∈ (0,2π)

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:
f(x) = x³ − 3x + 2, x ∈ [-2, 2]

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:
f (x) = (x − 2)(x − 7), x ∈ [3,11]

Show that the value in the conclusion of the mean value theorem for
\(f(x)=\frac{1}{x}\) on a closed interval of positive numbers [a, b] is \(\sqrt{ab} \)

Show that the value in the conclusion of the mean value theorem for
f(x) = Ax² + Bx + c on any interval [a, b] is \(\frac{a+b}{2}\)

A race car driver is racing at 20^th km. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours.

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

Does there exist a differentiable function f(x) such that f(0) = -1, f(2) = 4 and f'(x) ≤ 2 for all x. Justify you answer.

Show that there lies a point on the curve f(x) = x (x + 3) _e^{\(\frac{\pi}{2}\)}, -3 ≤ x ≤ 0 where tangent drawn is parallel to the x -axis.

A rectangular page is to contain 24 cm² of print. The margins at the top and bottom of the page are 1.5 cm and the margins at other sides of the page is 1 cm. What should be the dimensions of the page so that the area of the paper used is minimum.

Using mean value theorem prove that for, a > 0, b > 0, le^-a - e^-bl < la - bl.

Expand log(1+ x) as a Maclaurin’s series upto 4 non-zero terms for –1 < x ≤ 1.

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain1,80,000 sq. mtrs in order to provide enough grass for herds. No fencing is needed along the river. What is the length of the minimum needed fencing material

Expand tan x in ascending powers of x upto 5^th power for \(-\frac{\pi}{2} <x<\frac{\pi}{2}\)

Write the Taylor series expansion of \(\frac{1}{x}\) about x = 2 by finding the first three non-zero terms.

Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10 cm.

Prove that among all the rectangles of the given perimeter, the square has the maximum area.

Find the dimensions of the largest rectangle that can be inscribed in a semi circle of radius r cm.

A manufacturer wants to design an open box having a square base and a surface area of 108 sq. cm. Determine the dimensions of the box for the maximum volume.

The volume of a cylinder is given by the formula V = πr² h. Find the greatest and least values of V if r + h = 6.

A hollow cone with base radius a cm and, height b em is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is \(\frac { 4 }{ 9 } \) times volume of the cone.

Expand sin x in ascending powers x - \(\frac{\pi}{4}\) upto three non-zero terms.

Sketch the graphs of the following function. 
\(y=-\frac { 1 }{ 3 } \left( { x }^{ 3 }-3x+2 \right) \)

Sketch the graphs of the following function.
\(y=x\sqrt { 4-x } \)

Sketch the graphs of the following function.
\(y=\frac { { x }^{ 2 }+1 }{ { x }^{ 2 }-4 } \)

Sketch the graphs of the following function 
\(y=\frac { 1 }{ 1+{ e }^{ -x } } \)

Evaluate the following limit, if necessary use l’Hôpital Rule
\(​​​​​​\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ x }\)

For the function f{x) = 4x³ + 3x² - 6x + 1 find the intervals of monotonicity, local extrema, intervals of concavity and points of inflection.

Find the intervals of monotonicity and hence find the local extrema for the function \(f(x)=x^{\frac{2}{3}}\).

Determine the intervals of concavity of the curve f (x) = (x −1)³. (x − 5), x∈R and, points of inflection if any.

Sketch the curve y = f (x) = x² − x −6 .

Find the intervals of monotonicity and local extrema of the function \(f(x)=\frac{x }{1+x^{2}}\)

Find the local extrema of the function f(x) = 4x⁶ − 6x⁴

Prove that among all the rectangles of the given area square has the least perimeter.

Sketch the curve y = f (x) = x³−6x-9

Sketch the curve \(y=\frac { { x }^{ 2 }-3x }{ (x-1) } \)

Find the points on the unit circle x² + y² = 1 nearest and farthest from (1, 1).

A steel plant is capable of producing x tonnes per day of a low-grade steel and y tonnes per day of a high-grade steel, where \(y=\frac { 40-5x }{ 10-x } \). If the fixed market price of low-grade steel is half that of high-grade steel, then what should be optimal productions in low-grade steel and high-grade steel in order to have maximum receipts.

Sketch the graph of the function \(y=\frac { 3x }{ { x }^{ 2 }-1 } \)