#### Chapters

Chapter 2: Functions

Chapter 3: Binary Operations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Algebra of Matrices

Chapter 6: Determinants

Chapter 7: Adjoint and Inverse of a Matrix

Chapter 8: Solution of Simultaneous Linear Equations

Chapter 9: Continuity

Chapter 10: Differentiability

Chapter 11: Differentiation

Chapter 12: Higher Order Derivatives

Chapter 13: Derivative as a Rate Measurer

Chapter 14: Differentials, Errors and Approximations

Chapter 15: Mean Value Theorems

Chapter 16: Tangents and Normals

Chapter 17: Increasing and Decreasing Functions

Chapter 18: Maxima and Minima

Chapter 19: Indefinite Integrals

Chapter 20: Definite Integrals

Chapter 21: Areas of Bounded Regions

Chapter 22: Differential Equations

Chapter 23: Algebra of Vectors

Chapter 24: Scalar Or Dot Product

Chapter 25: Vector or Cross Product

Chapter 26: Scalar Triple Product

Chapter 27: Direction Cosines and Direction Ratios

Chapter 28: Straight Line in Space

Chapter 29: The Plane

Chapter 30: Linear programming

Chapter 31: Probability

Chapter 32: Mean and Variance of a Random Variable

Chapter 33: Binomial Distribution

## Chapter 16: Tangents and Normals

#### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.1 [Pages 10 - 11]

Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x^3} \text { at } x = 4\] ?

Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?

Find the slope of the tangent and the normal to the following curve at the indicted point *y* = *x*^{3} − *x* at *x* = 2 ?

Find the slope of the tangent and the normal to the following curve at the indicted point *y* = 2*x*^{2} + 3 sin *x* at *x* = 0 ?

Find the slope of the tangent and the normal to the following curve at the indicted point x = a (θ − sin θ), y = a(1 − cos θ) at θ = −π/2 ?

Find the slope of the tangent and the normal to the following curve at the indicted point *x* = *a* cos^{3} θ, *y* = *a* sin^{3} θ at θ = π/4 ?

Find the slope of the tangent and the normal to the following curve at the indicted point *x* = *a* (θ − sin θ), *y* = *a*(1 − cos θ) at θ = π/2 ?

Find the slope of the tangent and the normal to the following curve at the indicted point *y* = (sin 2*x* + cot *x* + 2)^{2} at *x* = π/2 ?

Find the slope of the tangent and the normal to the following curve at the indicted point *x*^{2} + 3*y** *+* **y*^{2} = 5 at (1, 1) ?

Find the slope of the tangent and the normal to the following curve at the indicted point *xy* = 6 at (1, 6) ?

Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?

If the tangent to the curve *y* = x^{3} + ax + b at (1, − 6) is parallel to the line x − y + 5 = 0, find a and b ?

Find a point on the curve *y* = *x*^{3} − 3*x* where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?

Find the points on the curve y = x^{3} − 2x^{2} − 2x at which the tangent lines are parallel to the line y = 2x− 3 ?

Find the points on the curve *y*^{2} = 2*x*^{3} at which the slope of the tangent is 3 ?

Find the points on the curve xy + 4 = 0 at which the tangents are inclined at an angle of 45° with the x-axis ?

Find the point on the curve y = x^{2} where the slope of the tangent is equal to the x-coordinate of the point ?

At what points on the circle x^{2} + y^{2} − 2x − 4y + 1 = 0, the tangent is parallel to x-axis?

At what point of the curve *y* = *x*^{2} does the tangent make an angle of 45° with the *x*-axis?

Find the points on the curve *y* = 3*x*^{2} − 9*x* + 8 at which the tangents are equally inclined with the axes ?

At what points on the curve *y* = 2*x*^{2} − *x* + 1 is the tangent parallel to the line *y* = 3*x* + 4?

Find the point on the curve *y* = 3*x*^{2} + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?

Find the points on the curve *x*^{2} + *y*^{2} = 13, the tangent at each one of which is parallel to the line 2*x* + 3*y* = 7 ?

Find the points on the curve 2a^{2}y = x^{3} − 3ax^{2} where the tangent is parallel to x-axis ?

At what points on the curve *y* = *x*^{2} − 4*x* + 5 is the tangent perpendicular to the line 2*y* + *x* = 7?

Find the points on the curve \[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the x-axis ?

Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?

Find the points on the curve x^{2} + y^{2} − 2x − 3 = 0 at which the tangents are parallel to the x-axis ?

Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to x-axis ?

Find the points on the curve \[\frac{x^2}{9} + \frac{y^2}{16} = 1\] at which the tangent is parallel to y-axis ?

Who that the tangents to the curve *y* = 7x^{3} + 11 at the points x = 2 and x = −2 are parallel ?

Find the points on the curve *y* = *x*^{3} where the slope of the tangent is equal to the x-coordinate of the point ?

#### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.2 [Pages 27 - 29]

Find the equation of the tangent to the curve \[\sqrt{x} + \sqrt{y} = a\] at the point \[\left( \frac{a^2}{4}, \frac{a^2}{4} \right)\] ?

Find the equation of the normal to *y* = 2*x*^{3} − *x*^{2} + 3 at (1, 4) ?

Find the equation of the tangent and the normal to the following curve at the indicated point *y *= *x*^{4} −* **bx*^{3} + 13*x*^{2} − 10*x* + 5 at (0, 5) ?

* *Find the equation of the tangent and the normal to the following curve at the indicated point y = x^{4} − 6x^{3} + 13x^{2} − 10x + 5 at x = 1 ?

Find the equation of the tangent and the normal to the following curve at the indicated point y = x^{2} at (0, 0) ?

Find the equation of the tangent and the normal to the following curve at the indicated point y = 2x^{2} − 3x − 1 at (1, −2) ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point y = x^{2} + 4x + 1 at x = 3 ?

Find the equation of the tangent and the normal to the following curve at the indicated point\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text{ at }\left( a\cos\theta, b\sin\theta \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( a\sec\theta, b\tan\theta \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point *y*^{2} = 4*a**x* at \[\left( \frac{a}{m^2}, \frac{2a}{m} \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[c^2 \left( x^2 + y^2 \right) = x^2 y^2 \text { at }\left( \frac{c}{\cos\theta}, \frac{c}{\sin\theta} \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point xy = c^{2} at \[\left( ct, \frac{c}{t} \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { at } \left( x_1 , y_1 \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?

Find the equation of the tangent and the normal to the following curve at the indicated point \[x^\frac{2}{3} + y^\frac{2}{3}\] = 2 at (1, 1) ?

Find the equation of the tangent and the normal to the following curve at the indicated point x^{2} = 4*y* at (2, 1) ?

Find the equation of the tangent and the normal to the following curve at the indicated point y^{2} = 4x at (1, 2) ?

Find the equation of the tangent and the normal to the following curve at the indicated point 4*x*^{2} + 9*y*^{2} = 36 at (3cos*θ*, 2sin*θ*) ?

Find the equation of the tangent and the normal to the following curve at the indicated point y^{2} = 4ax at (x_{1}, y_{1})?

Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?

Find the equation of the tangent to the curve *x* = θ + sin θ, *y* = 1 + cos θ at θ = π/4 ?

Find the equation of the tangent and the normal to the following curve at the indicated points x = θ + sinθ, y = 1 + cosθ at θ = \[\frac{\pi}{2}\] ?

Find the equation of the tangent and the normal to the following curve at the indicated points \[x = \frac{2 a t^2}{1 + t^2}, y = \frac{2 a t^3}{1 + t^2}\text { at } t = \frac{1}{2}\] ?

Find the equation of the tangent and the normal to the following curve at the indicated points x = at^{2}, y = 2at at t = 1 ?

Find the equation of the tangent and the normal to the following curve at the indicated points x = asect, y = btant at t ?

Find the equation of the tangent and the normal to the following curve at the indicated points x = a(θ + sinθ), y = a(1 − cosθ) at θ ?

Find the equation of the tangent and the normal to the following curve at the indicated points x = 3cosθ − cos^{3}θ, y = 3sinθ − sin^{3}*θ *?

Find the equation of the normal to the curve *x*^{2} + 2*y*^{2} − 4*x* − 6*y* + 8 = 0 at the point whose abscissa is 2 ?

Find the equation of the normal to the curve ay^{2} = x^{3} at the point (am^{2}, am^{3}) ?

The equation of the tangent at (2, 3) on the curve y^{2} = ax^{3} + b is y = 4x − 5. Find the values of a and b ?

Find the equation of the tangent line to the curve* y* = *x*^{2} + 4*x* − 16 which is parallel to the line 3*x* − *y* + 1 = 0 ?

Find an equation of normal line to the curve *y* = *x*^{3} + 2*x* + 6 which is parallel to the line *x*+ 14*y* + 4 = 0 ?

Determine the equation(s) of tangent (s) line to the curve y = 4x^{3} − 3x + 5 which are perpendicular to the line 9y + x + 3 = 0 ?

Find the equation of a normal to the curve y = x log_{e} x which is parallel to the line 2x − 2y + 3 = 0 ?

Find the equation of the tangent line to the curve y = x^{2} − 2x + 7 which is parallel to the line 2x − y + 9 = 0 ?

Find the equation of the tangent line to the curve *y* = *x*^{2} − 2*x* + 7 which perpendicular to the line 5*y* − 15*x* = 13. ?

Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?

Find the equations of all lines of slope zero and that are tangent to the curve \[y = \frac{1}{x^2 - 2x + 3}\] ?

Find the equation of the tangent to the curve \[y = \sqrt{3x - 2}\] which is parallel to the 4*x* − 2*y* + 5 = 0 ?

Find the equation of the tangent to the curve x^{2} + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?

Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?

Find the equation of the tangent to the curve *x* = sin 3*t*, *y* = cos 2*t* at

\[t = \frac{\pi}{4}\] ?

At what points will be tangents to the curve y = 2x^{3} − 15x^{2} + 36x − 21 be parallel to x-axis ? Also, find the equations of the tangents to the curve at these points ?

Find the equation of the tangents to the curve 3x^{2} – y^{2} = 8, which passes through the point (4/3, 0) ?

#### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals Exercise 16.3 [Pages 40 - 41]

Find the angle of intersection of the following curve y^{2} = x and x^{2} = y ?

Find the angle of intersection of the following curve y = x^{2} and x^{2} + y^{2} = 20 ?

Find the angle of intersection of the following curve 2y^{2}^{ }= x^{3} and y^{2} = 32x ?

Find the angle of intersection of the following curve x^{2}^{ }+ y^{2} − 4x − 1 = 0 and x^{2} + y^{2} − 2y − 9 = 0 ?

Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x^{2} + y^{2} = ab ?

Find the angle of intersection of the following curve x^{2} + 4y^{2} = 8 and x^{2} − 2y^{2} = 2 ?

Find the angle of intersection of the following curve x^{2} = 27y and y^{2} = 8x ?

Find the angle of intersection of the following curve x^{2} + y^{2} = 2x and y^{2} = x ?

Find the angle of intersection of the following curve *y* = 4 − *x*^{2} and *y* = *x*^{2} ?

Show that the following set of curve intersect orthogonally y = x^{3} and 6y = 7 − x^{2 }?

Show that the following set of curve intersect orthogonally x^{3} − 3xy^{2} = −2 and 3x^{2}y − y^{3} = 2 ?

Show that the following set of curve intersect orthogonally x^{2} + 4y^{2} = 8 and x^{2} − 2y^{2} = 4 ?

Show that the following curve intersect orthogonally at the indicated point *x*^{2} = 4*y* and 4*y* + *x*^{2} = 8 at (2, 1) ?

Show that the following curve intersect orthogonally at the indicated point x^{2} = y and x^{3} + 6y = 7 at (1, 1) ?

Show that the following curve intersect orthogonally at the indicated point y^{2} = 8x and 2x^{2} + y^{2} = 10 at \[\left( 1, 2\sqrt{2} \right)\] ?

Show that the curves 4x = y^{2} and 4xy = k cut at right angles, if k^{2} = 512 ?

Show that the curves 2x = y^{2} and 2xy = k cut at right angles, if k^{2} = 8 ?

Prove that the curves xy = 4 and x^{2} + y^{2} = 8 touch each other ?

Prove that the curves y^{2} = 4x and x^{2} + y^{2} - 6*x* + 1 = 0 touch each other at the point (1, 2) ?

Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { and } xy = c^2\] ?

Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?

Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?

If the straight line xcos \[\alpha\] +y sin \[\alpha\] = p touches the curve \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\] then prove that *a*^{2}cos^{2 \[\alpha\] }\[-\] b^{2}sin^{2 }\[\alpha\] = p^{2 }?

#### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 41 - 42]

Find the point on the curve y = x^{2} − 2x + 3, where the tangent is parallel to x-axis ?

Find the slope of the tangent to the curve x = t^{2} + 3t − 8, y = 2t^{2} − 2t − 5 at t = 2 ?

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to x-axis, then write the value of \[\frac{dy}{dx}\] ?

Write the value of \[\frac{dy}{dx}\] , if the normal to the curve *y* = f(x) at (x, y) is parallel to y-axis ?

If the tangent to a curve at a point (*x*, *y*) is equally inclined to the coordinates axes then write the value of \[\frac{dy}{dx}\] ?

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of \[\frac{dx}{dy}\] ?

Find the slope of the normal at the point '*t*' on the curve \[x = \frac{1}{t}, y = t\] ?

Write the coordinates of the point on the curve y^{2} = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?

Write the angle made by the tangent to the curve x = e^{t} cos t, y = e^{t} sin t at \[t = \frac{\pi}{4}\] with the *x*-axis ?

Write the equation of the normal to the curve *y* = *x* + sin *x* cos *x* at \[x = \frac{\pi}{2}\] ?

Find the coordinates of the point on the curve y^{2} = 3 − 4x where tangent is parallel to the line 2x + y− 2 = 0 ?

Write the equation on the tangent to the curve y = x^{2} − x + 2 at the point where it crosses the y-axis ?

Write the angle between the curves y^{2} = 4x and x^{2} = 2y − 3 at the point (1, 2) ?

Write the angle between the curves y = e^{−x} and y = e^{x} at their point of intersections ?

Write the slope of the normal to the curve \[y = \frac{1}{x}\] at the point \[\left( 3, \frac{1}{3} \right)\] ?

Write the coordinates of the point at which the tangent to the curve y = 2x^{2} − x + 1 is parallel to the line y = 3x + 9 ?

Write the equation of the normal to the curve y = cos x at (0, 1) ?

Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?

#### RD Sharma solutions for Class 12 Maths Chapter 16 Tangents and Normals [Pages 42 - 44]

The equation to the normal to the curve *y* = sin x at (0, 0) is ___________ .

*x*= 0*y*= 0*x*+*y*= 0*x*−*y*= 0

The equation of the normal to the curve y = x + sin x cos x at x = `π/2` is ___________ .

*x*= 2*x*= π*x*+ π = 02

*x*= π

The equation of the normal to the curve y = x(2 − x) at the point (2, 0) is ________________ .

x − 2y = 2

x − 2y + 2 = 0

2x + y = 4

2x + y − 4 = 0

The point on the curve y^{2} = x where tangent makes 45° angle with x-axis is ______________ .

(1/2, 1/4)

(1/4, 1/2)

(4, 2)

(1, 1)

If the tangent to the curve *x* = a t^{2}, y = 2 at is perpendicular to x-axis, then its point of contact is _____________ .

(a, a)

(0, a)

(0, 0)

(a, 0)

The point on the curve y = x^{2} − 3x + 2 where tangent is perpendicular to y = x is ________________ .

(0, 2)

(1, 0)

(−1, 6)

(2, −2)

The point on the curve y^{2} = x where tangent makes 45° angle with x-axis is ____________________ .

(1/2, 1/4)

(1/4, 1/2)

(4, 2)

(1, 1)

The point at the curve *y* = 12*x* − *x*^{2} where the slope of the tangent is zero will be _____________ .

(0, 0)

(2, 16)

(3, 9)

none of these

The angle between the curves y^{2} = x and x^{2}^{ }= y at (1, 1) is ______________ .

\[\tan^{- 1} \frac{4}{3}\]

\[\tan^{- 1} \frac{3}{4}\]

90°

45°

The equation of the normal to the curve 3x^{2} − y^{2} = 8 which is parallel to x + 3y = 8 is ____________ .

x + 3y = 8

x + 3y + 8 = 0

x + 3y ± 8 = 0

x + 3y = 0

The equations of tangent at those points where the curve *y* = *x*^{2} − 3*x* + 2 meets *x*-axis are _______________ .

*x*−*y*+ 2 = 0 =*x*−*y*− 1*x*+*y*− 1 = 0 =*x*−*y*− 2*x*−*y*− 1 = 0 =*x*−*y**x*−*y*= 0 =*x*+*y*

The slope of the tangent to the curve x = t^{2} + 3 t − 8, y = 2t^{2} − 2t − 5 at point (2, −1) is ________________ .

22/7

6/7

`-6`

none of these

At what point the slope of the tangent to the curve *x*^{2} + *y*^{2} − 2*x* − 3 = 0 is zero

(3, 0), (−1, 0)

(3, 0), (1, 2)

(−1, 0), (1, 2)

(1, 2), (1, −2)

The angle of intersection of the curves xy = a^{2} and x^{2} − y^{2} = 2a^{2}^{ }is ______________ .

0°

45°

90°

none of these

If the curve ay + x^{2} = 7 and x^{3} = y cut orthogonally at (1, 1), then a is equal to _____________ .

1

`-6`

6

0

If the line y = x touches the curve y = x^{2} + bx + c at a point (1, 1) then _____________ .

b = 1, c = 2

b = −1, c = 1

b = 2, c = 1

b = −2, c = 1

The slope of the tangent to the curve x = 3t^{2} + 1, y = t^{3} −1 at x = 1 is ___________ .

1/2

0

`-2`

∞

The curves y = ae^{x} and y = be^{−x} cut orthogonally, if ___________ .

a = b

a = −b

ab = 1

ab = 2

The equation of the normal to the curve *x* = a cos^{3} θ, y = a sin^{3} θ at the point θ = π/4 is __________ .

x = 0

y = 0

x = y

x + y = a

If the curves y = 2 e^{x} and y = ae^{−x} intersect orthogonally, then a = _____________ .

1/2

−1/2

2

2e

^{2}

The point on the curve y = 6x − x^{2} at which the tangent to the curve is inclined at π/4 to the line x + y= 0 is __________ .

(−3, −27)

(3, 9)

(7/2, 35/4)

(0, 0)

The angle of intersection of the parabolas y^{2} = 4 ax and x^{2} = 4ay at the origin is ____________ .

π/6

π/3

π/2

π/4

The angle of intersection of the curves y = 2 sin^{2} x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .

π/4

π/2

π/3

none of these

Any tangent to the curve y = 2x^{7} + 3x + 5 __________________ .

is parallel to x-axis

is parallel to y-axis

makes an acute angle with x-axis

makes an obtuse angle with x-axis

The point on the curve 9*y*^{2} = *x*^{3}, where the normal to the curve makes equal intercepts with the axes is

(a) \[\left( 4, \frac{8}{3} \right)\]

(b) \[\left( - 4, \frac{8}{3} \right)\]

(c) \[\left( 4, - \frac{8}{3} \right)\]

(d) none of these

The slope of the tangent to the curve x = t^{2} + 3t − 8, y = 2t^{2} − 2t − 5 at the point (2, −1) is _____________ .

\[\frac{22}{7}\]

\[\frac{6}{7}\]

\[\frac{7}{6}\]

\[- \frac{6}{7}\]

The line y = mx + 1 is a tangent to the curve y^{2} = 4x, if the value of m is ________________ .

1

2

3

`1/2`

The normal at the point (1, 1) on the curve 2y + x^{2} = 3 is _____________ .

x + y = 0

x − y = 0

x + y + 1 = 0

x − y = 1

The normal to the curve x^{2} = 4y passing through (1, 2) is _____________ .

x + y = 3

x − y = 3

x + y = 1

x − y = 1

none of these

## Chapter 16: Tangents and Normals

## RD Sharma solutions for Class 12 Maths chapter 16 - Tangents and Normals

RD Sharma solutions for Class 12 Maths chapter 16 (Tangents and Normals) 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 CBSE Class 12 Maths solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Maths chapter 16 Tangents and Normals are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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