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

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

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

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

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

1. Define odd and even function, with example.

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

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

    (b) Prove that the  does not exist

1(a) If f(x) = x2 then find .

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

2. Show that the series  Converges to -1.

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

    (b) Prove that  the does not exist.

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. 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.

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

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).

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

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

3. Test the convergence of the series 

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

3. Does the following series converge? 

3. Test the convergence of the series 

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. Evaluate: 

3. Test the convergence of the series 

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

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

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

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

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

4. Find the eccentricity of the curve 2x² + y² = 4.

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

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]

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

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

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

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

2(b)Sketch the curve[5]

5. Find the angle between the planes 3x − 6y − 2z = 15 and 2x + y − 2z = 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 point where the line X = 8/3 + 2t, y = -2t, z = 1 + t intersects the plane 3x + 2y + 6z = 6.

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

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

6. Evaluate 

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

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

6. Evaluate the integral 

6. Evaluate 

6. Evaluate 

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

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

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

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

7. Find  and if f(x,y) = x² + y²

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

7. Evaluate 

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

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. Find and if f(x, y) = 10 − x² − y².

7. Find  and  if f(x, y) = ye².

7. Evaluate the limit 

4. (a) If  does exist? Justify. 

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

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

8. Evaluate 

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

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

8. Prove that 

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

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

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

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

4. (a) Evaluate[5]

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

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

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

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

9. Show that 

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

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

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

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

10.Solve 

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

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

10.Solve 

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

10.Find the general solution of the equation 

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 

5. If  and , find gof and gog.

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

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

6. Use continuity to evaluate the limit , 

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

11. State and prove Rolle ’s Theorem.

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

f(x) =  on the interval [-1,1].

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

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

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

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

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

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

12. Test if the following series converges 

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

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

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

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

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

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

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.

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

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

13. Find the Cartesian equation of the polar equation 

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

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

9. Evaluate 

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

8. Sketch the curve y = x³ + x

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) = x³ - x² - 6x + 2 in [0, 3].

9. Determine whether the integer  is convergent or divergent .

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,

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

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.

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.

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. Evaluate 

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 it 

15. Obtain the general solution of 

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

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

15. Find a general solution of the differential equation 

15. Find the solution of the equation 

15. Solve 

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

15. Obtain the general solution of 

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 general solution of 

15. Find the particular integral of the equation 

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

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

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

12. Determine whether the series  converges or diverges. 

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. 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

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

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

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

16. Find the area of the region enclosed by the parabola y = 2 – x² and the line y = -x.

OR

Evaluate the integrals

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. 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.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.

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).

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

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

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. 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. 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.

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

17. Find the curvature of the helix 

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

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

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

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