# Mathematics I (Calculus) - Unit Wise Questions

1. (a) A function is defined by f(x) = |x| , calculate f(-3), f(4), and sketch the graph.

1. Define a relation and a function from a set into another set. Give suitable example.

1. Define odd and even function, with example.

1. (a) A function is defined by , calculate f(-1),f(3), and sketch the graph.(5)

(b) Prove that the does not exist

1. Define one-to-one and onto functions with suitable examples.

1. Verify the men value theorem for the function f(x) = √x(x − 1) in the interval [0, 1].

1. Find the length of the curve y = x^{3/2} from x=0 to x =4.

1. If f(x) = sin x and g(x) = -x/2. Find f(f(x)) and g(f(x)).

1. If f(x) = (x − 1) + x,then prove that f(x). f(1 − x) = 1

1. If f(x) = x + 2 and g(x) = x^{3} − 3 find g(f(3)).

1(a) If f(x) = x2 then find .

2. Show that the area under the arch of the curve y = sin x is.

2. Define critical point .Find the critical point of f(x)=x^{2}.

2. Find the critical points of the function f(x) = x^{3/2} (x-4).

2. Obtain the area between two curves y = sec2x and y = sin x from x = 0 to x = π/4.

2. Show by integral test that the series converges if p>1.

2. Find the length of the curve for 0 ≤ x ≤ 1.

2. Show that the series converges by using integral test.

1(b) Dry air is moving upward. If the ground temperature is 20^{0} and the temperature at a height of 1km is 10^{0} C, express the temperature T in ^{0}C as a function of the height h (in kilometers), assuming that a linear model is appropriate. (b)Draw the graph of the function in part(a). What does the slope represent? (c) What is the temperature at a height of 2km?(5)

2. Define critical point. Find the critical point of f(x) = 2x2.

(b) Prove that the does not exist.

2. Show that the series Converges to -1.

2. (a) Find the domain and sketch the graph of the function f(x) = x^{2} - 6x .

3. Does the following series converge?

3. Test the convergence of the series

3. Test the convergence of the series By comparison test.

3. Evaluate:

3. Investigate the convergence of the series

3. Test the convergence of the series

3. Test the convergence of the series

(b) Estimate the area between the curve y = x^{2 }and the lines y = 1 and y = 2.

3. Evaluate

1(c). Find the equation of the tangent to the parabola y = x^{2} + x + 1 at (0, 1)

4. Find the equation of the parabola with vertex at the origin and focus at (0,2).

4. Find the polar equation of the circle (x+2)^{2} + y^{2} = 4.

4. Find the equation of the parabola with vertex at the origin and directrix at x= 7.

3. (a) Find the Maclaurin series for cos x and prove that it represents cos x for all x.

4. Find the equation of the parabola with vertex at the origin and directrix at y=2

4. Find the eccentricity of the curve 2x^{2} + y^{2} = 4.

2(a)A farmer has 2000 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimentions of the field that has the largest area?[5]

4. Obtain the semi-major axis ,semi-minor axis,foci,vertices

4. Find the focus and the directrix of the parabola y^{2} = 10x.

4. Find the foci, vertices, center of the ellipse

5. Find the angle between the planes 3x − 6y − 2z = 15 and 2x + y − 2z = 5

5. Find the point where the line X = 8/3 + 2t, y = -2t, z = 1 + t intersects the plane 3x + 2y + 6z = 6.

5. Find the angle between the planes x − 2y − 2z = 5 and 5x − 2y − z = 0

5. Find a vector parallel to the line of intersection of the planes 3x + 6y – 2z = 5.

3. (a) Find the Maclaurin series for e^{x} and prove that it represents e^{x} for all x.

5. Find the area of the parallelogram where vertices are A(0,0), B(7,3), C(9,8) and D(2,5).

5. Find the angle between the planes 3x − 6y − 2z = 7 and 2x + y − 2z = 5

5. Find the angle between the vectors 2i+j+k and -4i+3j+k.

5. Find the equation for the plane through (-3,0,7) perpendicular to

2(b)Sketch the curve[5]

(b) Define initial value problem. Solve that initial value problem of y' + 2y = 3, y(0) = 1.

6. Evaluate

6. Find the velocity and acceleration of a particle whose position is

(c) Find the volume of a sphere of a radius* a .*

6. Evaluate

6. Find a spherical coordinate equation for the sphere x^{2} + y^{2} + (z-1)^{2} = 1.

6. Evaluate

6. Obtain the area of the region R bounded by y=x and y= x^{2} in the first quadratic .

6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder 4x^{2} + 4y^{2 }= 9 in cylindrical coordinates.

3(a)Show that the converges and diverges .[2]

6. Evaluate the integral

7. Find and if f(x,y) = x^{2} + y^{2}

7. Show that the function Is continuous at every point in the plane except the origin.

7. Evaluate the limit

4. (a) If does exist? Justify.

7. Find and if f(x, y) = ye^{2}.

7. Calculate for f(x,y) = 1 – 6x^{2}y, R : 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.

(c) Find the volume of a sphere of radius r.

(b) If f(x, y) = xy/(x^{2} + y^{2}), does f(x, y) exist, as (x, y) → (0, 0)?[3]

7. Evaluate

7. Find the area of the region R bounded by y = x and y = x^{2} in the first quadrant by using double integrals.

7. Find and if f(x, y) = 10 − x^{2} − y^{2}.

8. Define Jacobian determinant for x = g(u, v, w), y = h(u, v, w), z = k(u, v, w).

8. Prove that

8. Find the equation for the tangent plane to the surfaces Z = f(x, y) = g − x^{2} − y^{2} at the point (1,2,3).

4(b) Calculate for f(x, y) = 100 - 6x^{2}y and

3(c) A particle moves in a straight line and has acceleration given by *a(t) = 6t ^{2} + 1*. Its initial velocity is

*4m/sec*and its initial displacement is

*s(0) = 5cm*. Find its position function

*s(t)*.[5]

8. Find the Jacobean j(u,v,w) if x=u+v, y=2 u,z=3w.

8. Using partial derivatives ,find if 2xy + tany − 4y^{2} = 0.

8. Define Jacobian determinant for X = g(u, v, w) ,y = h(u, v, w), z = k(u, v, w).

8. Find if ω = x^{2} + y – z + sin t and x + y = t.

8. Evaluate

9. Solve the partial differential equation p + q = x.

9. Show that y = x^{2} + 5 is the solution of

9. Show that y = ax^{2} + b is the solution of xy’’ + y’ = 0.

4. (a) Evaluate[5]

9. What do you mean by local extreme points of f(x,y)? Illustrate the concept by graphs.

9. Show that y = c_{1}xe^{−2x} + c_{2}e^{−2x} is the solution of y′′ + y′ − 2y = 0.

(b) Calculate ∫ ∫ f(x, y)dA for f(x, y) = 100 − 6x^{2}y and R: 0 ≤ x ≤ 2, −1 ≤ y ≤ 1.

9. Show that

9. Verify that the partial differential equation is satisfied by .

9. Find the extreme values of f(x,y) = x^{2}+ y^{2}.

5. If f(x) = and g(x) = , find fog and fof.

10. Define partial differential equations of the first index with suitable examples.

10.Solve

10.Find the general solution of the equation

10.Solve

10.Find and at (1,2) of f(x, y) = x^{2} + 2xy + 5.

10.Define partial differential equations of the second order with suitable examples.

10. Find the general integral of the linear partial differential equation z(xp – yq) = z^{2} – x^{2} .

10.Solve

4(b) Find the Maclaurin's series for cos x and prove that it represents cos x for all x.[5]

6. Define continuity on an interval. Show that the function is continuous on the interval [ -1,1] .

5. If and , find gof and gog.

11. State and prove Rolle ’s Theorem.

11. State and prove mean value theorem for definite integral.

11. State Rolles’s theorem and verify it for the functionf(x) = sinx in [0, π].

11. Verify Rolles’s theorem for the function f(x) = x^{2 }− 5x + 7 in the interval [2,3].

11. Verify Rolles’s theorem for f(x) = x^{2}, x ∈ [−1,1].

11. Verify Rolle’s theorem for f(x) = x3, x ∈ [-3,3].

6. Use continuity to evaluate the limit ,

7. Verify Mean value theorem of f(x) = x^{3} - 3x + 2 for [-1, 2].

11. State the mean value theorem for a differentiable function and verify it for the function

f(x) = on the interval [-1,1].

11. State Rolle’s Theorem for a differential function. Support with examples that the hypothesis of theorem are essential to hold the theorem.

12. Find the Taylor series and Taylor polynomials generated by the function f(x) = cos x at x = 0.

12. Find the Taylors series and the Taylor polynomials generated by f(x) = e^{x} at x = 0.

5. If f(x) = x^{2} - 1, *g(x) = 2x + 1*, find *fog* and *gof *and domain of *fog*.

12. Test if the following series converges

8. Stating with x_{1} = 2, find the third approximation x_{3} to the root of the equation x^{3} - 2x - 5 = 0.

12. Find the Taylors series expression of f(x) = sin x at x = 0.

12. Find the Taylors series expression of f(x) = cos θ at x = 1.

7. Verify Mean value theorem of f(x) = x^{3 }− 3x + 3 for [−1,2].

12. Find the length of the cardioid r = 1 + cosθ.

12. Find the area of the region that lies in the plane enclosed by the cardioid r = 2(i + cosθ).

12. Find the Taylor series expansion of the case at ex, at x=0.

13. What do you mean by principle unit normal vector? Find unit tangent vector and principle unit vector for the circular motion

13. Define unit tangent vector of a differentiable curve. Find the unit tangent vector of the curve r(t) = (cos t + t sin t)i + (sin t – t cos t)j, t > 0.

9. Evaluate

8. Sketch the curve y = x^{3} + x

13. Find the length of the cardioid r = 1 – cosθ.

13. Obtain the polar equations for circles through the origin centered on the x and y axis and radius a.

13. Find the length of the cardioids r = 1 + cosθ.

13. Find a Cartesian equivalent of the polar equation r cos (θ-π/3) = 3.

13. Obtain the polar equations for circles through the origin centered on x and y axis ,with radius a.

13. Find the Cartesian equation of the polar equation

6. Define continuity of a function at a point *x = a*. Show that the function f(x) = is continuous on the interval[1, -1].

14. Show that the function is continuous at every point except the origin.

7. State Rolle's theorem and verify the Rolle's theorem for* f(x) = x ^{3} - x^{2} - 6x + 2* in [0, 3].

14. Find the gradient vector of f(x,y) at a pointP(x_{0}, y_{0}).Find an equation for the tangent to the ellipse x^{2} + 4y^{2} = 4 at point (−2,1).

14. Evaluate

14. Show that the function is continuous at every point except the origin .

14. Evaluate it

14. Define partial derivative of a function f(x,y) with respect to x at the point (x_{0}y_{0}).State Euler’s theorem ,verify if it for the function .f(x, y) = x^{2} + 5xy + sinx + 7e^{x},

9. Determine whether the integer is convergent or divergent .

10. Find the volume of the resulting solid which is enclosed by the curve y = x and y = x^{2 }is rotated about the x-axis.

14. Define the partial derivative of f(x,y) at a point (x_{0}, y_{0}) with respect to all variables. Find the derivative of f(x,y) = xe^{y} = cos(x, y) at the point (2, 0) in the direction of A = 3i – 4j.

14. What do you mean by critical point of a function f(x,y) in a region? Find local extreme values of the function f(x,y) = xy – x^{2}– y^{2} – 2x – 2y + 4.

15. Find a general solution of the differential equation

15. Solve

11. Find the solution of y'' + 4y' + 4 = 0.

8. Find the third approximation x_{3} to the root of the equation *f(x) = x ^{3} - 2x - 7*, setting

*x1 = 2*.

15. Find the center of mass of a solid of constant density δ, bounded below by the disk: x^{2} + y^{2} = 4 in the plane z = 0 and above by the paraboid z = 4 – x^{2} – y^{2}.

15. Find a particular integral of the equation = 2y – x^{2}

15. Find the solution of the equation

15. Obtain the general solution of

15. Obtain the general solution of

15. Find the particular integral of the equation

15. Find the general solution of

12. Determine whether the series converges or diverges.

10.Find the length of the arc of the semicubical parabola y^{2} = x^{3} between the point(1,1) and (4,8).

9. Find the derivatives of *r(t) = (1 + t ^{2})i - te^{-t}j +*

*sin 2tk*and find the unit tangent vector at

*t=0*.

16. Graph the function f(x) = -x^{3} + 12x + 5 for -3 ≤x ≤ 3.

16. Graph the function y = x^{4/3}– 4x^{1/3}

16. State Lagranhes’s mean value theorem and verify the theorem for x = x^{3} − x^{2} − 5x + 3in [0,4].

**Or**

Investigates the convergence of the integrals

16. Find the area bounded on right by the line y=x-2 on the left by the parabola x=y^{2} and below by the x-axis

**Or**

What is an improper integral? Evaluate

16.Evaluate the integrals and determine whether they converge or diverge

**OR**

Find the area bounded on the parabola y = 2 – x2 and the line y = -x.

16. Graph the function

16. Find the area of the region enclosed by the parabola y = 2 – x^{2} and the line y = -x.

**OR**

Evaluate the integrals

16. Find the area of the region in the first quadrant that is bounded above by y = √x and below by the x-axis and the line y = x – 2.

OR

Investigate the convergence of the integrals

16. Find the area of the region bounded by x = 2y^{2}. , x = 0 and y = 3.

Or

Investigates the convergence of the integrals

10. Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x^{2}.

13. If a = (4, 0, 3) and b = (-2, 1, 5) find |a|, the vector a - b and 2a + b.

11. Find the solution of y" + 6y′ + 9 = 0, y(0) = 2, y(0) = 1.

17. Calculate the curvature and torsion for the helix r(t) = (a cos t)i + (a sin t)j + btk,a,b ≥ 0, a^{2} + b^{2} ≠ 0.

17. What do you mean by Taylor’s polynomial of order n? Obtain Taylor’s polynomial and

Taylor’s series generated by the function f(x) =cos x at x =0.

17. What is mean by maclaurin series? Obtain the maclaurin series for the function

17. Define a curvature of a space curve. Find the curvature for the helix r(t) = (a cost)i + (a sint)j + btk(a,b ≥ 0, a^{2} + b^{2} ≠ 0).

11. Solve: y" + y' = 0, y(0) = 5, y(π/4) = 3

14. Find and if z is defined as a function of x and y by the equation x^{3} + y^{3} + z^{3} + 6xyz = 1.

17. Define curvature of a curve .find that the curvature of a helix

17. Define curvature of a curve .Show that the curvature of a (a) straight line on zero and (b) a circle of a radius a is l/a .

17. Find the curvature of the helix

17. Find the torsion ,normal and curvature for the space curve

12. Show that the series converges.

15. Find the extreme values of the function f(x, y) = x^{2} + 2y^{2} on the circle x^{2} + y^{2} = 1.

18. Find the volume of the region enclosed by the surface z = x^{2}+ 3y^{2} and z = 8 – x^{2}– y^{2}.

18. Find the volume of the region D enclosed by the surfaces z = x^{2} + 3y^{2} and z = 8 – x^{2} – y^{2}.

12. Test the convergence of the series

18.Find the volume of the region D enclosed by the surfaces z = x^{2}+ 3y^{2} and z = 8 – x^{2} – y^{2}.

18.Evaluate the double integral by applying the transformation and integrating over an appropriate region in the uv-plane.

18.Evaluate

18.Find the volume enclosed between the surfaces z = x^{2} + 3y^{2} and z = 8 – x^{2} – y^{2}

18.Find the area enclosed by r^{2} = 2a^{2} cos 2θ

18.Find the volume enclosed between the surfaces Z = x^{2} + 3y^{2} and Z = 8 − x^{2} − y^{2}

13. Find a vector perpendicular to the plane that passes through the points:p(1, 4, 6), Q(-2, 5, -1) and R(1. -1, 1)

19. Find the maximum and minimum of the function f(x, y) = x^{3} + y^{3} − 12x + 20.

**OR**

Find the Point on the ellipse x^{2 }+ 2y^{2 }= 1 where f(x, y) = xy has its extreme values.

19. Define maximum and minimum of a function at a point .Final the local maximum and local minimum of the function f(x, y) = 2xy − 5x^{2} − 2y^{2} + 4x + 4y − 4.

19. Find the local maximum , minimum and saddles point of 6x^{2} − 2x^{3} + 3y^{2} + 6xy.

**OR**

Find the greatest and smallest values that the function f(x,y) =xy takes on the ellipse

19. Find the extreme values of the function F(x,y) = xy –x^{2} –y^{2} -2x -2y + 4

**OR**

Find the extreme values of f(x,y) = xy subject to g(x,y) = x^{2} + y^{2} – 10 = 0.

19. Obtain the absolute maximum and minimum values of the function. f(x,y) = 2 + 2x + 2y – x^{2}– y^{2} on the triangular plate in the first quadrant bounded by lines x = 0, y = 0, y = 9 – x.

**OR**

Evaluate the integral

20. Define initial boundary values problems .Derive the heat equation or wave equation in one dimension .

13. Define cross product of two vectors .if a=i+3j +4k and b-= 2i+7j=5k, find the vector a × b and b × a.

19. Find the extreme values of Z = x^{3} − y^{3} − 2xy + 6.

**OR**

Find the extreme value of function F(x, y) = xy takes on the ellipse

19. Find the absolute maximum and minimum values of f(x,y) = 2 + 2x + 2y – x^{2}– y^{2} on the triangular plate in the first quadrant bounded by lines x = 0, y = 0 and x + y =9.

**OR**

Find the points on the curve xy^{2}= 54 nearest to the origin. How are the Lagrange multipliers defined?

19. Find the maximum and minimum values of the function f(x,y) = 3x + 4y on the circle x^{2} +y^{2} = 1.

**OR**

State the conditions of second derivative test for local extreme values. Find the local extreme values of the function f(x,y) = x^{2 }+ xy + y^{2 }+ 3x – 3y + 4.

20. Define second order partial differential equation. Define initial boundary value problem. Derive the heat equation or wave equation in one dimension.

20. Define one-dimensional wave equation and one-dimensional heat equations with initial conditions. Derive solution of any of them.

20. Show that the solution of the wave equation and deduce the result if the velocity is zero.

**OR**

Find a particular integral of the equation (D^{2} − D^{1}) = A cos(lx + my) where A, l, m are constants.

14. Define limit of a function . find

14. Find the partial derivative of f(x, y) = x^{3} + 2x^{3}y^{3} - 3y^{2} + x + y, at (2,1).

20. Derive D’ Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.

20. Find the solution of the equation

**Or**

Find the particular integral of the equation

20. Define the wave equation by the modeling of vibrating string.

20. Define second order partial differential equation .What is initial boundary values problem ?Solve :u_{t} = u_{xx} = u_{tt} = u_{xx}

15. Find the extreme value of f(x, y) = y^{2} − x^{2} .

15. Find the local maximum and minimum values, saddle points of f(x,y) = x^{4} + y^{4} - 4xy + 1.

2. (a) Find the derivative of f(x) = √x and to state the domain of f^{℩}

11. State and prove the mean value theorem for a differential function.

14. What is meant by direction derivative in the plain? Obtain the derivative of the function f(x,y) = x^{2}+ xy at P(1, 2) in the direction of the unit vector

19. Find the maximum and the minimum values of f(x, y) = 2xy – 2y^{2}– 5x^{2} + 4x – 4. Also find the saddle point if it exists.

**OR**

Evaluate the integral

(b) Estimate the area between the curve y^{2} = x and the lines x=0 and x=2.

6. Find the area enclosed by the curve r^{2} = 4cos2θ.

13. Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.

(b) Define initial value problem. Solve that initial value problem of y' + 5y = 1, y(0) = 2.

3. Test the convergence of p – series for p > 1.

4. (a) For what value of x does the series converge?

17. Define Taylor’s polynomial of order n. Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = e^{x} at x = 0.

5. Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).

7. Obtain the values of and at the point (4, -5) if f(x,y) = x^{2}+ 3xy + y -1.

8. Using partial derivatives , find if x^{2} + cos y – y^{2}= 0.

1. Verify Rolle’s theorem for the function on the interval [-3, 3].

4. Find the eccentricity of the hyperbola 9x^{2} – 16y^{2} = 144.

9. Find the partial differential equation of the function (x – a)^{2} + (y – b)^{2} + z^{2}= c^{2}.

10. Solve the partial differential equation x^{2}p + q = z^{2 }.

12. Find the length of the Asteroid x = cos^{3}t, y = sin^{3}t for 0 ≤ t ≥ 2π.

18. Obtain the centroid and the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x^{2}.

20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.

**OR**

Find the particular integral of the equation (D^{2} – D^{1})z =2y-x^{2} where