Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Applications of Differential Calculus [Latest edition]

## Chapter 7: Applications of Differential Calculus

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10
Exercise 7.1 [Pages 8 - 9]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.1 [Pages 8 - 9]

Exercise 7.1 | Q 1. (i) | Page 8

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

Exercise 7.1 | Q 1. (ii) | Page 8

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

Exercise 7.1 | Q 2. (i) | Page 8

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?

Exercise 7.1 | Q 2. (ii) | Page 8

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?

Exercise 7.1 | Q 2. (iii) | Page 8

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?

Exercise 7.1 | Q 3. (i) | Page 8

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?

Exercise 7.1 | Q 3. (ii) | Page 8

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

Exercise 7.1 | Q 3. (iii) | Page 8

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

Exercise 7.1 | Q 4 | Page 8

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

Exercise 7.1 | Q 5 | Page 8

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

Exercise 7.1 | Q 6 | Page 8

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?

Exercise 7.1 | Q 7 | Page 8

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?

Exercise 7.1 | Q 8 | Page 8

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?

Exercise 7.1 | Q 9. (i) | Page 8

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?

Exercise 7.1 | Q 9. (ii) | Page 8

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?

Exercise 7.1 | Q 10 | Page 9

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?

Exercise 7.2 [Pages 14 - 15]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.2 [Pages 14 - 15]

Exercise 7.2 | Q 1. (i) | Page 14

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

y = x4 + 2x2 – x at x = 1

Exercise 7.2 | Q 1. (ii) | Page 14

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

Exercise 7.2 | Q 2 | Page 14

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

Exercise 7.2 | Q 3 | Page 14

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

Exercise 7.2 | Q 4 | Page 14

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

Exercise 7.2 | Q 5. (i) | Page 15

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

y = x2 – x4 at (1, 0)

Exercise 7.2 | Q 5. (ii) | Page 15

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

y = x4 + 2ex at (0, 2)

Exercise 7.2 | Q 5. (iii) | Page 15

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)

Exercise 7.2 | Q 5. (iv) | Page 15

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

Exercise 7.2 | Q 6 | Page 15

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

Exercise 7.2 | Q 7 | Page 15

Find the equations of the tangents to the curve y = - (x + 1)/(x - 1) which are parallel to the line x + 2y = 6

Exercise 7.2 | Q 8 | Page 15

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

Exercise 7.2 | Q 9 | Page 15

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

Exercise 7.2 | Q 10 | Page 15

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

Exercise 7.3 [Pages 21 - 22]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.3 [Pages 21 - 22]

Exercise 7.3 | Q 1. (i) | Page 21

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

f(x) = |1/x|, x ∈ [- 1, 1]

Exercise 7.3 | Q 1. (ii) | Page 21

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

f(x) = tan x, x ∈ [0, π]

Exercise 7.3 | Q 1. (iii) | Page 21

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]

Exercise 7.3 | Q 2. (i) | Page 21

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]

Exercise 7.3 | Q 2. (ii) | Page 21

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]

Exercise 7.3 | Q 2. (iii) | Page 21

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]

Exercise 7.3 | Q 3. (i) | Page 21

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]

Exercise 7.3 | Q 3. (ii) | Page 21

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]

Exercise 7.3 | Q 4. (i) | Page 21

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]

Exercise 7.3 | Q 4. (ii) | Page 21

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]

Exercise 7.3 | Q 5. (i) | Page 21

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")

Exercise 7.3 | Q 5. (ii) | Page 21

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

Exercise 7.3 | Q 6 | Page 21

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

Exercise 7.3 | Q 7 | Page 21

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

Exercise 7.3 | Q 8 | Page 22

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

Exercise 7.3 | Q 9 | Page 22

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

Exercise 7.3 | Q 10 | Page 22

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

Exercise 7.4 [Page 25]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.4 [Page 25]

Exercise 7.4 | Q 1. (i) | Page 25

Write the Maclaurin series expansion of the following functions:

ex

Exercise 7.4 | Q 1. (ii) | Page 25

Write the Maclaurin series expansion of the following functions:

sin x

Exercise 7.4 | Q 1. (iii) | Page 25

Write the Maclaurin series expansion of the following functions:

cos x

Exercise 7.4 | Q 1. (iii) | Page 25

Write the Maclaurin series expansion of the following functions:

log(1 – x); – 1 ≤ x ≤ 1

Exercise 7.4 | Q 1. (v) | Page 25

Write the Maclaurin series expansion of the following functions:

tan–1 (x); – 1 ≤ x ≤ 1

Exercise 7.4 | Q 1. (vi) | Page 25

Write the Maclaurin series expansion of the following functions:

cos2x

Exercise 7.4 | Q 2 | Page 25

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

Exercise 7.4 | Q 3 | Page 25

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

Exercise 7.4 | Q 4 | Page 25

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

Exercise 7.5 [Pages 31 - 32]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.5 [Pages 31 - 32]

Exercise 7.5 | Q 1 | Page 31

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

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

Exercise 7.5 | Q 2 | Page 31

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

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

Exercise 7.5 | Q 3 | Page 31

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

lim_(x -> oo) x/logx

Exercise 7.5 | Q 4 | Page 31

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

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

Exercise 7.5 | Q 5 | Page 31

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

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

Exercise 7.5 | Q 6 | Page 31

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

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

Exercise 7.5 | Q 7 | Page 31

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

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

Exercise 7.5 | Q 8 | Page 31

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

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

Exercise 7.5 | Q 9 | Page 31

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

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

Exercise 7.5 | Q 10 | Page 32

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

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

Exercise 7.5 | Q 11 | Page 32

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

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

Exercise 7.5 | Q 12 | Page 32

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 = A0ert

Exercise 7.6 [Page 40]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.6 [Page 40]

Exercise 7.6 | Q 1. (i) | Page 40

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

f(x) = x2 – 12x + 10; [1, 2]

Exercise 7.6 | Q 1. (ii) | Page 40

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

f(x) = 3x4 – 4x3 ; [– 1, 2]

Exercise 7.6 | Q 1. (iii) | Page 40

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

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

Exercise 7.6 | Q 1. (iv) | Page 40

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

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

Exercise 7.6 | Q 2. (i) | Page 40

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

f(x) = 2x3 + 3x2 – 12x

Exercise 7.6 | Q 2. (ii) | Page 40

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

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

Exercise 7.6 | Q 2. (iii) | Page 40

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

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

Exercise 7.6 | Q 2. (iv) | Page 40

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

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

Exercise 7.6 | Q 2. (v) | Page 40

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)

Exercise 7.7 [Page 44]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.7 [Page 44]

Exercise 7.7 | Q 1. (i) | Page 44

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

f(x) = x(x – 4)3

Exercise 7.7 | Q 1. (ii) | Page 44

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

f(x) = sin x + cos x, 0 < x < 2π

Exercise 7.7 | Q 1. (iii) | Page 44

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

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

Exercise 7.7 | Q 2. (i) | Page 44

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

f(x) = – 3x5 + 5x3

Exercise 7.7 | Q 2. (ii) | Page 44

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

f(x) = x log x

Exercise 7.7 | Q 2. (iii) | Page 44

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

f(x) = x2 e–2x

Exercise 7.7 | Q 3 | Page 44

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

Exercise 7.8 [Page 47]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.8 [Page 47]

Exercise 7.8 | Q 1 | Page 47

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

Exercise 7.8 | Q 2 | Page 47

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

Exercise 7.8 | Q 3 | Page 47

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

Exercise 7.8 | Q 4 | Page 47

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?

Exercise 7.8 | Q 5 | Page 47

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?

Exercise 7.8 | Q 6 | Page 47

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?

Exercise 7.8 | Q 7 | Page 47

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

Exercise 7.8 | Q 8 | Page 47

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

Exercise 7.8 | Q 9 | Page 47

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

Exercise 7.8 | Q 10 | Page 47

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

Exercise 7.8 | Q 11 | Page 47

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

Exercise 7.8 | Q 12 | Page 47

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

Exercise 7.9 [Page 53]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.9 [Page 53]

Exercise 7.9 | Q 1. (i) | Page 53

Find the asymptotes of the following curves:

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

Exercise 7.9 | Q 1. (ii) | Page 53

Find the asymptotes of the following curves:

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

Exercise 7.9 | Q 1. (iii) | Page 53

Find the asymptotes of the following curves:

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

Exercise 7.9 | Q 1. (iv) | Page 53

Find the asymptotes of the following curves:

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

Exercise 7.9 | Q 1. (v) | Page 53

Find the asymptotes of the following curves:

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

Exercise 7.9 | Q 2. (i) | Page 53

Sketch the graphs of the following functions

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

Exercise 7.9 | Q 2. (ii) | Page 53

Sketch the graphs of the following functions:

y = xsqrt(4 - x)

Exercise 7.9 | Q 2. (iii) | Page 53

Sketch the graphs of the following functions:

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

Exercise 7.9 | Q 2. (iv) | Page 53

Sketch the graphs of the following functions:

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

Exercise 7.10 [Pages 54 - 55]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 7 Applications of Differential Calculus Exercise 7.10 [Pages 54 - 55]

Exercise 7.10 | Q 1 | Page 54

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

• 3 cm/s

• 2 cm/s

• 1 cm/s

• 1/2 cm/s

Exercise 7.10 | Q 2 | Page 54

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

• 3/25 radians/sec

• 4/25 radians/sec

• 1/5 radians/sec

• 1/3 radians/sec

Exercise 7.10 | Q 3 | Page 54

Choose the correct alternative:

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

• t = 0

• t = 1/3

• t = 1

• t = 3

Exercise 7.10 | Q 4 | Page 54

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

• 2

• 2.5

• 3

• 3.5

Exercise 7.10 | Q 5 | Page 54

Choose the correct alternative:

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

• (4, 11)

• (4, – 11)

• (– 4, 11)

• (– 4, – 11)

Exercise 7.10 | Q 6 | Page 54

Choose the correct alternative:

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

• – 8

• – 4

• – 2

• 0

Exercise 7.10 | Q 7 | Page 54

Choose the correct alternative:

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

• - 4sqrt(3)

• – 4

• - sqrt(3)/12

• 4sqrt(3)

Exercise 7.10 | Q 8 | Page 54

Choose the correct alternative:

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

• y = 0

• y = +-  sqrt(3)

• y = 1/2

• y = +-  3

Exercise 7.10 | Q 9 | Page 54

Choose the correct alternative:

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

• tan^-1  3/4

• tan^-1  (4/3)

• pi/2

• pi/4

Exercise 7.10 | Q 10 | Page 54

Choose the correct alternative:

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

• 0

• 1

• 2

• oo

Exercise 7.10 | Q 11 | Page 55

Choose the correct alternative:

The function sin4x + cos4x is increasing in the interval

• [(5pi)/8, (3pi)/4]

• [pi/2, (5pi)/8]

• [pi/4, pi/2]

• [0, pi/4]

Exercise 7.10 | Q 12 | Page 55

Choose the correct alternative:

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

• 1

• sqrt(2)

• 3/2

• 2

Exercise 7.10 | Q 13 | Page 55

Choose the correct alternative:

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

• 2

• 2.5

• 3

• 3.5

Exercise 7.10 | Q 14 | Page 55

Choose the correct alternative:

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

• 0

• 3

• 6

• 9

Exercise 7.10 | Q 15 | Page 55

Choose the correct alternative:

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

• x = pi/4

• x = pi/2

• x = pi

• x = (3pi)/2

Exercise 7.10 | Q 16 | Page 55

Choose the correct alternative:

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

• 1/"e"

• 1/(2"e")

• 1/"e"^2

• 4/"e"^4

Exercise 7.10 | Q 17 | Page 55

Choose the correct alternative:

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

• (2, 0)

• (sqrt(5), 1)

• (3, sqrt(5))

• (sqrt(13), - sqrt(13))

Exercise 7.10 | Q 18 | Page 55

Choose the correct alternative:

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

• 100

• 25sqrt(7)

• 28

• 24sqrt(14)`

Exercise 7.10 | Q 19 | Page 55

Choose the correct alternative:

The curve y = ax4 + bx2 with ab > 0

• has no horizontal tangent

• is concave up

• is concave down

• has no points of inflection

Exercise 7.10 | Q 20 | Page 55

Choose the correct alternative:

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

• (0, 0)

• (0, 1)

• (0, 1)

• (1, 1)

## Chapter 7: Applications of Differential Calculus

Exercise 7.1Exercise 7.2Exercise 7.3Exercise 7.4Exercise 7.5Exercise 7.6Exercise 7.7Exercise 7.8Exercise 7.9Exercise 7.10

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 - Applications of Differential Calculus

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 (Applications of Differential Calculus) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Tamil Nadu Board Samacheer Kalvi textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 7 Applications of Differential Calculus are Applications of Differential Calculus, Meaning of Derivatives, Mean Value Theorem, Series Expansions, Indeterminate Forms, Applications of First Derivative, Applications of Second Derivative, Applications in Optimization, Symmetry and Asymptotes, Sketching of Curves.

Using Tamil Nadu Board Samacheer Kalvi Class 12th solutions Applications of Differential Calculus exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Tamil Nadu Board Samacheer Kalvi Solutions are important questions that can be asked in the final exam. Maximum students of Tamil Nadu Board of Secondary Education Class 12th prefer Tamil Nadu Board Samacheer Kalvi Textbook Solutions to score more in exam.

Get the free view of chapter 7 Applications of Differential Calculus Class 12th extra questions for Class 12th Mathematics Volume 1 and 2 Answers Guide and can use Shaalaa.com to keep it handy for your exam preparation