Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 7 - Applications of Differential Calculus [Latest edition]

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. Find the average velocity between 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. Find the instantaneous velocities at t = 3 and t = 6 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. How long does the camera fall before 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. What is the instantaneous velocity of the camera when it hits the ground?

A particle moves along a line according to the law s(t) = 2t³ – 9t² + 12t – 4, where t ≥ 0. At what times the particle changes direction?

A particle moves along a line according to the law s(t) = 2t³ – 9t² + 12t – 4, where t ≥ 0. Find the total distance travelled by the particle in the first 4 seconds

A particle moves along a line according to the law s(t) = 2t³ – 9t² + 12t – 4, where t ≥ 0. Find the particle’s acceleration each time the velocity is zero

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

If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = `sqrt(3x)` then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 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?

A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shoreline. How fast is the beam moving along the shoreline 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. How fast is the top of the ladder moving down the wall?

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, 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?

Find the slope of the tangent to the following curves at the respective given points.

y = x⁴ + 2x² – x at x = 1

Find the slope of the tangent to the following curves at the respective given points.

x = a cos³t, y = b sin³t at t = `pi/2`

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

Find the points on 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 tangent and normal to the following curves at the given points on the curve

y = x² – x⁴ at (1, 0)

Find the tangent and normal to the following curves at the given points on the curve

y = x⁴ + 2e^x at (0, 2)

Find the tangent and normal to the following curves at the given points on the curve

y = x sin x at `(pi/2, pi/2)`

Find the tangent and normal to the following curves at the given points on the curve

x = cos t, y = 2 sin²t at t = `pi/2`

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 = `- (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 andy = 2 sin 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

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x) = |1/x|, x ∈ [- 1, 1]`

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x)` = tan x, x ∈ [0, π]

Explain why Rolle’s theorem is not applicable to the following functions in the respective intervals

`f(x)` = x – 2 log x, x ∈ [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]

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) = (x^2 - 2x)/(x + 2), x ∈ [-1, 6]`

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) = sqrt(x) - x/3, x ∈ [0, 9]`

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:

`f(x) = (x + 1)/x, x ∈ [-1, 2]`

Explain why Lagrange’s mean value theorem is not applicable to the following functions in the respective intervals:

`f(x) = |3x + 1|, x ∈ [-1, 3]`

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^3 - 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) = 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) = "A"x^2 + "B"x + "C"` on any interval [a, b] is `("a" + "b")/2`

A race car driver is kilometer stone 20. 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) ≤ 1 for 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 your answer

Show that there lies a point on the curve `f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0` where tangent drawn is parallel to the x-axis

Using Mean Value Theorem prove that for, a > 0, b > 0, |e^–a – e^–b| < |a – b|

Write the Maclaurin series expansion of the following functions:

e^x

Write the Maclaurin series expansion of the following functions:

sin x

Write the Maclaurin series expansion of the following functions:

cos x

Write the Maclaurin series expansion of the following functions:

log(1 – x); – 1 ≤ x ≤ 1

Write the Maclaurin series expansion of the following functions:

tan^–1 (x); – 1 ≤ x ≤ 1

Write the Maclaurin series expansion of the following functions:

cos²x

Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0

Expand sin x in ascending powers `x - pi/4` upto three non-zero terms

Expand the polynomial f(x) = x² – 3x + 2 in power of x – 1

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 0) (1 - cosx)/x^2`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x - oo) (2x^2 - 3)/(x^2 -5x + 3)`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> oo) x/logx`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> x/2) secx/tanx`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> oo) "e"^-x sqrt(x)`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 0) (1/sinx - 1/x)`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 1^+) (2/(x^2 - 1) - x/(x - 1))`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 0^+) x^x`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> oo) (1 + 1/x)^x`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> pi/2) (sin x)^tanx`

Evaluate the following limits, if necessary use l’Hôpital Rule:

`lim_(x -> 0^+) (cos x)^(1/x^2)`

Evaluate the following limits, if necessary use l’Hôpital Rule:

If an initial amount A₀ of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = `"A"_0 (1 + "r"/"n")^"nt"`. If the interest is compounded continuously, (that is as n → ∞), show that the amount after t years is A = A₀e^rt 

Find the absolute extrema of the following functions on the given closed interval.

f(x) = x² – 12x + 10; [1, 2]

Find the absolute extrema of the following functions on the given closed interval.

f(x) = 3x⁴ – 4x³ ; [– 1, 2]

Find the absolute extrema of the following functions on the given closed interval.

f(x) = `6x^(4/3) - 3x^(1/3) ; [-1, 1]`

Find the absolute extrema of the following functions on the given closed interval.

f(x) = `2 cos x + sin 2x; [0, pi/2]`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = 2x³ + 3x² – 12x

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `x/(x - 5)`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `"e"^x/(1 - "e"^x)`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = `x^3/3 - log x`

Find the intervals of monotonicities and hence find the local extremum for the following functions:

f(x) = sin x cos x + 5, x ∈ (0, 2π)

Find intervals of concavity and points of inflexion for the following functions:

f(x) = x(x – 4)³ 

Find intervals of concavity and points of inflection for the following functions:

f(x) = sin x + cos x, 0 < x < 2π

Find intervals of concavity and points of inflection for the following functions:

f(x) = `1/2 ("e"^x - "e"^-x)`

Find the local extrema for the following functions using second derivative test:

f(x) = – 3x⁵ + 5x³

Find the local extrema for the following functions using second derivative test:

f(x) = x log x

Find the local extrema for the following functions using second derivative test:

f(x) = x² e^–2x 

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

Find two positive numbers whose sum is 12 and their product is maximum

Find two positive numbers whose product is 20 and their sum is minimum

Find the smallest possible value of x² + y² given that x + y = 10

A garden is to be laid out in a rectangular area and protected by a wire fence. What is the largest possible area of the fenced garden with 40 meters of wire?

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 the other sides of the page are 1 cm. What should be the dimensions’ of the page so that the area of the paper used is minimum?

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 1,80,000 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?

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 = `pi"r"^2"h"`. Find the greatest and least values of V if r + h = 6

A hollow cone with a base radius of a cm and’ height of b cm is placed on a table. Show that) the volume of the largest cylinder that can be hidden underneath is `4/9` times the volume of the cone

Find the asymptotes of the following curves:

f(x) = `x^2/(x^2 - 1)`

Find the asymptotes of the following curves:

f(x) = `x^2/(x + 1)`

Find the asymptotes of the following curves:

f(x) = `(x^2 - 6x - 1)/(x + 3)`

Find the asymptotes of the following curves:

f(x) = `(x^2 - 6x - 1)/(x + 3)`

Find the asymptotes of the following curves:

f(x) = `(x^2 + 6x - 4)/(3x - 6)`

Sketch the graphs of the following functions

y = `- 1/3 (x^3 - 3x + 2)`

Sketch the graphs of the following functions:

y = `xsqrt(4 - x)`

Sketch the graphs of the following functions:

y = `(x^2 + 1)/(x^2 - 4)`

Sketch the graphs of the following functions:

y = `1/(1 + "e"^-x)`

Choose the correct alternative:

The volume of a sphere is increasing in volume at the rate of 3π cm³/ sec. The rate of change of its radius when radius is `1/2` cm

Choose the correct alternative:

A balloon rises straight up at 10 m/s. An observer is 40 m away from the spot where the balloon left the ground. The rate of change of the balloon’s angle of elevation in radian per second when the balloon is 30 metres above the ground

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The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t² – 2t – 8. The time at which the particle is at rest is

Choose the correct alternative:

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

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

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The abscissa of the point on the curve f(x) = `sqrt(8 - 2x)` at which the slope of the tangent is – 0.25?

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The slope of the line normal to the curve f(x) = 2 cos 4x at x = `pi/12` is

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

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Angle between y² = x and x² = y at the origin is

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The value of the limit `lim_(x -> 0) (cot x - 1/x)` is

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The function sin⁴x + cos⁴x is increasing in the interval

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The number given by the Rolle’s theorem for the function x³ – 3x², x ∈ [0, 3] is

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The number given by the Mean value theorem for the function `1/x`, x ∈ [1, 9] is

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The minimum value of the function `|3 - x| + 9` is

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The maximum slope of the tangent to the curve y = e^x sin x, x ∈ [0, 2π] is at

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The maximum value of the function x² e^-2x, x > 0 is

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

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The maximum value of the product of two positive numbers, when their sum of the squares is 200, is

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The curve y = ax⁴ + bx² with ab > 0

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