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.

    (b) Prove that the  does not exist

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

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

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

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

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

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

1. Define odd and even function, with example.

1(a) If f(x) = x2 then find .

1. If f(x) = x + 2 and g(x) = x³ − 3 find g(f(3)).

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

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

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

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

    (b) Prove that  the does not exist.

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

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

2. Show that the series  Converges to -1.

1(b) Dry air is moving upward. If the ground temperature is 20⁰ and the temperature at a height of 1km is 10⁰ C, express the temperature T in ⁰C 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. 

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

2. (a) Find the domain and sketch the graph of the function f(x) = x² - 6x . 

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

3. Evaluate 

3. Investigate the convergence of the series 

1(c). Find the equation of the tangent to the parabola y = x² + x + 1 at (0, 1)

    (b) Estimate the area between the curve y = x² and the lines y = 1 and y = 2. 

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

3. Test the convergence of the series 

3. Does the following series converge? 

3. Test the convergence of the series 

3. Evaluate: 

3. Test the convergence of the series 

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. Find the focus and the directrix of the parabola y² = 10x.

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

4. Find the eccentricity of the curve 2x² + y² = 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 focus at (0,2).

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

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

4. Find the polar equation of the circle (x+2)² + y² = 4.

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

5. Find a vector parallel to the line of intersection of the planes 3x + 6y – 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 vectors 2i+j+k and -4i+3j+k.

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

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

2(b)Sketch the curve[5]

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 = 15 and 2x + y − 2z = 5

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

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

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

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

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

6. Evaluate 

6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder 4x² + 4y² = 9 in cylindrical coordinates.

6. Obtain the area of the region R bounded by y=x and y= x² in the first quadratic .

6. Evaluate the integral 

6. Evaluate 

6. Evaluate 

6. Find a spherical coordinate equation for the sphere x² + y² + (z-1)² = 1.

7. Find  and if f(x,y) = x² + y²

7. Find and if f(x, y) = 10 − x² − y².

7. Find the area of the region R bounded by y = x and y = x² in the first quadrant by using double integrals.

7. Evaluate 

7. Calculate  for f(x,y) = 1 – 6x²y, R : 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.

4. (a) If  does exist? Justify. 

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

(b) If f(x, y) = xy/(x² + y²), does f(x, y) exist, as (x, y) → (0, 0)?[3]

7. Evaluate the limit 

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

7. Find  and  if f(x, y) = ye².

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

8. Evaluate 

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

3(c) A particle moves in a straight line and has acceleration given by a(t) = 6t² + 1. Its initial velocity is 4m/sec and its initial displacement is s(0) = 5cm. Find its position function s(t).[5]

8. Find  if ω = x² + y – z + sin t and x + y = t.

8. Prove that 

8. Using partial derivatives ,find if 2xy + tany − 4y² = 0.

8. Find the equation for the tangent plane to the surfaces Z = f(x, y) = g − x² − y² at the point (1,2,3).

4(b) Calculate  for  f(x, y) = 100 - 6x²y and  

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

9. Show that y = c₁xe^−2x + c₂e^−2x is the solution of y′′ + y′ − 2y = 0.

9. Show that 

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

9. Show that y = x² + 5 is the solution of 

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

9. Find the extreme values of f(x,y) = x²+ y².

9. Show that y = ax² + b is the solution of xy’’ + y’ = 0.

    (b) Calculate ∫ ∫ f(x, y)dA for f(x, y) = 100 − 6x²y and R: 0 ≤ x ≤ 2, −1 ≤ y ≤ 1.

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

4. (a) Evaluate[5]

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

10.Solve 

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

10. Find the general integral of the linear partial differential equation z(xp – yq) = z² – x² .

10.Solve 

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

10.Find  and  at (1,2) of f(x, y) = x² + 2xy + 5.

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

10.Find the general solution of the equation 

10.Solve 

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

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

11. Verify Rolles’s theorem for f(x) = x², x ∈ [−1,1].

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

7. Verify Mean value theorem of f(x) = x³ - 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].

6. Use continuity to evaluate the limit , 

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

11. State and prove Rolle ’s Theorem.

11. Verify Rolles’s theorem for the function f(x) = x² − 5x + 7 in the interval [2,3].

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

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

12. Test if the following series converges 

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

5. If f(x) = x² - 1, g(x) = 2x + 1, find fog and gof and domain of fog.

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

8. Stating with x₁ = 2, find the third approximation x₃ to the root of the equation x³ - 2x - 5 = 0. 

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

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

7. Verify Mean value theorem of f(x) = x³ − 3x + 3 for [−1,2].

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

13. Find the Cartesian equation of the polar equation 

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

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

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

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.

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

9. Evaluate 

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

8. Sketch the curve y = x³ + x

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

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

14. Define partial derivative of a function f(x,y) with respect to x at the point (x₀y₀).State Euler’s theorem ,verify if it for the function .f(x, y) = x² + 5xy + sinx + 7e^x,

7. State Rolle's theorem and verify the Rolle's theorem for f(x) = x³ - x² - 6x + 2 in [0, 3].

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

9. Determine whether the integer  is convergent or divergent .

14. Find the gradient vector of f(x,y) at a pointP(x₀, y₀).Find an equation for the tangent to the ellipse x² + 4y² = 4 at point (−2,1).

14. Evaluate 

14. Evaluate it 

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 – x²– y² – 2x – 2y + 4.

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

14. Define the partial derivative of f(x,y) at a point (x₀, y₀) 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.

10. Find the volume of the resulting solid which is enclosed by the curve y = x and y = x² is rotated about the x-axis.

15. Obtain the general solution of 

15. Find a particular integral of the equation  = 2y – x²

15. Solve 

8. Find the third approximation x₃ to the root of the equation f(x) = x³ - 2x - 7, setting x1 = 2.

15. Obtain the general solution of 

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

15. Find the center of mass of a solid of constant density δ, bounded below by the disk: x² + y² = 4 in the plane z = 0 and above by the paraboid z = 4 – x² – y².

15. Find the solution of the equation 

15. Find a general solution of the differential equation 

15. Find the particular integral of the equation 

15. Find the general solution of 

16. Graph the function f(x) = -x³ + 12x + 5 for -3 ≤x ≤ 3.

10.Find the length of the arc of the semicubical parabola y² = x³ between the point(1,1) and (4,8).

9. Find the derivatives of r(t) = (1 + t²)i - te^-tj + sin 2tk and find the unit tangent vector at t=0.

12. Determine whether the series  converges or diverges. 

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.

16. Find the area bounded on right by the line y=x-2 on the left by the parabola x=y² and below by the x-axis

Or

What is an improper integral? Evaluate 

16. State Lagranhes’s mean value theorem and verify the theorem for x = x³ − x² − 5x + 3in [0,4].

Or

Investigates the convergence of the integrals 

16. Find the area of the region enclosed by the parabola y = 2 – x² 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. Graph the function y = x^4/3– 4x^1/3

16. Graph the function 

16. Find the area of the region bounded by x = 2y². , x = 0 and y = 3.

Or

Investigates the convergence of the integrals 

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.

10. Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x².

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

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. 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, a² + b² ≠ 0).

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

17. Calculate the curvature and torsion for the helix r(t) = (a cos t)i + (a sin t)j + btk,a,b ≥ 0, a² + b² ≠ 0.

14. Find  and  if z is defined as a function of x and y by the equation x³ + y³ + z³ + 6xyz = 1. 

17. Find the curvature of the helix 

18.Find the area enclosed by r² = 2a² cos 2θ

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