PAPER 1 CALCULUSASSIGNMENT GROUP 1

SECTION 1

(i) Inverse Functions

1. Discuss the restrictions on the domain and range in the definition of the inverse trigonometric functions.

2. Find the inverse (if it exists) of the function h(x) = (2x – 6)/(3x + 3). 3. Determine whether the pair of functions f(x) = (2/3)x + 2; g(x) = (3/2)x + 3, are

inverses of each other. 4. Suppose that α is an acute angle of a right triangle where sin α = (s2 – t2)/(s2 + t2) (s

> t > 0). Show that α = tan -1((s2 – t2)/(2st)). 5. Sketch the graph of f(x) = (1 – x2)1/2, on (-1,1). 6. A painting 3ft high is hung on a wall in such a way that its lower edge is 7 ft

above the floor. An observer whose eyes are 5ft above the floor stands x ft away from the wall. Express the angle θ subtended by the painting as a function of x.

7. Show that if f -1 exists, it is unique. 8. Draw the line y = x. then use symmetry with respect to the line y = x to add the

graph of f -1 to your sketch. Identify the domain and range of f -1.

200 400 6 00 8 00 10 0 0

0.0 0 0 02

0.0 0 0 04

0.0 0 0 06

0.0 0 0 08

y 1

x2 1

9. Graph the function f(x) = 1/x. What symmetry does the graph have? Show that f is its own inverse.

10. For f(x) = x2 + 1, x ≥ 0, find a formula for f -1. 11. If f(x) = x5, find f -1(x) and identify the domain and range of f -1. Show that

.

12. Find the inverse, if it exists, of the function f = {(4, 3),(5, 6),(0, 7)}. 13. Use the horizontal line test to determine whether y = 2x has an inverse. If f-1

exists, draw its inverse. 14. Find the inverse of the function defined as .

15. Find the inverse of the function defined as .

16. Find the inverse of the function defined as .

17. Find the inverse of the function .

18. How would you restrict the domain of the function so that a unique inverse exists?

19.State ‘True’ or ‘False’. Justify your answer:a. Inflection point must be on the graph of a function. b. implies that the limit of f(x) is a large number.

c. The second derivative of f(x) = x - is .

d. The graph of the function y = will have at most 1 inflection point.

e. The range of sech x is the (0, ∞). 20.

(ii) Hyperbolic Functions

1. If , find the values of remaining five hyperbolic functions.

2. Rewrite in terms of exponentials and simplify the result.

3. Compute the integral .

4. Show that .

5. Show that .

6. Express the inverse hyperbolic cosine functions in terms of the logarithmic function.

7.

(iii) Higher Order Derivatives

1. If , show that .

2. , show that

.

3. Find the nth derivatives, given that

(i) .

(ii) . 4. Find the equation of the tangent line to the ellipse 25x2 + y2 = 109.

(iv) Leibniz Rule and its Applications

1. Prove that:

(i) .

(ii) .

2. Find the nth derivative of . 3. Show that the nth derivative of the differential equation

is

.

4. If show that

. 5.

(v) Concavity and Inflection Points

1. Identify the inflection points and local maxima and minima of the function

. Identify the intervals on which the functions are concave up

and concave down. 2. Graph the equation . Include the coordinates of any local

extreme points and inflection points. 3. is the first derivative of a continuous function . Find

and sketch the general shape of the graph of f.

4. Sketch a smooth connected curve with

5. Can anything be said about the graph of a function y = f(x) that has a continuous second derivative that is never zero? Give reasons for your answer.

6. What is the relationship between concavity, point of inflection and the second derivative?

7. For the function , (a) Find where the function is increasing and decreasing. (b) Find the critical points and identify each as a relative maximum, relative

minimum or neither. (c) Find the second order critical numbers and tell where the graph is concave up

and where it is concave down. (d) Sketch the graph.

8. Determine the intervals of increase and decrease and concavity for the given function , and then use those intervals to help you sketch its graph.

9. Use the first order test to classify the critical points as a relative

minimum, a relative maximum or neither for the function .

10. Use the second order derivative test to classify x = -9, x = 1 as a relative

minimum, a relative maximum or neither for .

11. Sketch the graph of a function with the following properties: f’(x) > 0 when x < 1, f’(x) < 0 when x > 1, f’’(x) > 0 for x < 1 and f’’(x) > 0 when x > 1.

12. Give an elementary proof that is positive and

increasing. 13. What are critical numbers? Discuss the importance of critical numbers in curve

sketching. 14. Either show that the following statement is generally true or find a

counterexample. “If f and g are concave up on the interval I, then so is f + g”.

15. Find the points of inflection on the following curve: . 16. Find the inflection points for the function f(x) = 3x5 – 5x3 + 2. 17. Sketch the graphs of f(x) = 2x3 – 3x2 – 12x + 1 and its derivative on the same

graph. Do you observe anything special on the graph?18. Find the local extrema of f(x) = x5 – 5x. 19. For the function f(x) = x9/5 – x, find first and second critical points and then sketch

the graph of the function.

20.Calculate the inflection points of y = (x + 1)/(x2 + 1).21.Do detailed graphing for f(x) = x - 3x1/3. 22.

(vi) Asymptotes

1. Find the limit of the function as and as .

2. Find .

3. Find the limit of the rational function as and as

.

4. Find .

5. Sketch the graph of a function that satisfies

. 6. Find a function that satisfies the given conditions and sketch its graph:

.

7. Suppose an odd function is known to be increasing on the interval . What

can be said of its behavior on the interval ?

8. Evaluate .

9. Find all the vertical and horizontal asymptotes of the graph of .

Find where the graph is rising and where it is falling, determine concavity and locate all critical points and points of inflection. Finally, sketch the graph. Find if there is any cusp or vertical tangent.

10. According to Einstein’s special theory of relativity, the mass of a body is modeled

by the expression where m0 is the mass of the body at rest in relation

to the observer, m is the mass of the body when it moves with speed v in relation to the observer and c is the speed of light. Sketch the graph of m as a function of v. What happens as ?

11. Find the asymptotes of the following curve:

(i) x3 + 2x2y – xy2 – 2y3 + xy – y2 – 1 = 0. (ii) (y – a)2(x2 – a2) = x4 + a4.

12. Find the position and nature of the double points on the following curves:(i) y2 – x(x – a)2 = 0. (ii) y6 – 3a4y3 + 2x3 – 3ax2 + a3 = 0.

13. Find . 14. Find whether f(x) = |x3 – 2| has vertical tangents or cusps.

15.Compute   .

16.Find whether the function has vertical tangent or cusp.

17.

(vii) L’Hopital’s Rule

1. Use l’Hopital’s rule to find .

2. Find .

3. Find .

4. Which one is correct and which one is wrong? Give reasons for your answer.

(a) .

(b) .

5.An incorrect use of l’hopital’s rule is illustrated. Explain what is wrong and find the

correct value of the limit. .

6.Find .

7.Find .

8.Find A so that = 5.

9.Use l’hopital’s rule to determine all horizontal asymptotes to the graph of the function

.

10. Find constants a and b so that .

11. Evaluate:

(i) .

(ii) .

12.

(viii) Optimization Procedure

1. What is the largest possible area for a right triangle whose hypotenuse is 5cm long? 2. An 1125 ft3 open top rectangular tank with a square base x ft on a side and y ft deep is

to be built with its top flush with the ground to catch runoff water. The costs associated with the tank involve not only the material from which the tank is made but also an excavation charge proportional to the product xy. If the cost is

, what values of x and y will minimize it?

3. A right triangle whose hypotenuse is long is revolved about one of its legs to generate a right circular cone. Find the radius, height and volume of the cone of greatest volume that can be made this way?

4. Suppose that at any given time t (sec) the current i (amp) in an alternating current is I = 2 cos t + 2 sin t. what is the peak current for this circuit (largest magnitude)?

5. Find the points on the curve nearest the point (c, 0) (a) if c ≥ 0 (b) if c < 0. 6. Find the amount of medicine to which the body is most sensitive by finding the value

of M that maximizes the derivative dR/dM, where and C is a

constant. 7. Find the dimension of the right circular cylinder of largest volume that can be

inscribed in a right circular cone of radius R and altitude H. 8.One end of a cantilever beam of length L is built into a wall and other end is supported

by a single post. The deflection, or sag, of the beam at a point located x units from the built-in end is modeled by the formula where k is a positive constant. Where does the maximum deflection occur on the beam?

9.It is known that water expand and contracts according to its temperature. Physical experiments suggest that an amount of water that occupies 1 litre at C will occupy V(T) = 1 – 6.42 × 10-5T + 8.51 × 10-6T2 – 6.79 × 10-8T3 litres when the temperature is

T C. at what temperature is V(T) minimized? How is this result related to the fact that ice is formed only at the upper levels of a lake during winter?

10.

(ix) Fermat’s Principle of Optics and Snell’s Law

1.Light from a source A is reflected by a plane mirror to a receiver at point B. Show that for the light to obey Fermat’s principle, the angle of incidence must be equal to angle of reflection.

2.What is Fermat’s principle of optics? 3.

(x) Applications in Business, Economics and Life Sciences.

1.Suppose that the dollar cost of producing a washing machine is

. (a) Find the average cost per machine of producing the first 100 washing machines.(b) Find the marginal cost when 100 washing machines are produced. (c) Show that the marginal cost when 100 washing machines are produced is

approximately the cost of producing one more washing machine after the first 100 have been made by calculating the later cost directly.

2.An open-top box is to be made by cutting small congruent squares from the corners of a 12 by 12 in sheet of tin and bending up the sides. How large should the squares cut from the corners be to make the box hold as much as possible.

3.Let c(t) denote the concentration in blood at time t of a drug injected into the body intramuscularly. It was observed that the concentration may be modeled by:

, t ≥ 0. Where a, b and k are positive constants that

depend on the drug. At what time does the largest concentration occur? What

happens to the concentration at t → ∞?

4.If and , where C is the cost of producing x units

of a particular commodity and selling price p when x units are produced, determine the level of production that maximizes profit.

5.Suppose the total revenue (in Rs.) from the sale of x units of a certain commodity is R(x) = –2x2 + 68x – 128. (a) At what level of sales is average revenue per unit equal to the marginal revenue?(b) Verify that the average revenue is increasing if the level of sales is less than the

level in part a and decreasing if the level of sales is greater than the level in part a. (c) On the same set of axes, graph the relevant portions of the average and marginal

revenue functions.6.Suppose that the demand function for a certain commodity is expressed as

for 0 ≤ x ≤ 120, where x is the number of items sold.

a. Find the total revenue function explicitly and use its first derivative to determine the price at which revenue is maximized.

b. Graph the relevant portions of the demand and revenue function. 7.According to a certain logistic model, the world’s population (in billion) t years after

1960 is modeled by the function .

a. If this model is correct, at what rate will the world’s population be increasing with respect to time in the year 2010? At what percentage rate will it be increasing at this time?

b. Sketch the graph of P. What feature on the graph corresponds to the time when the population is growing most rapidly? What happens to P(t) as t → +∞?

8.An important quantity in economic analysis is elasticity of demand, defined by

, where x is the number of units of a commodity demanded when the

price is Rs. p per unit. Show that .

1. (a) Show that .

(b) Use Leibniz rule to find the nth derivative of .

(c) Identify the inflection points and local maxima and minima of the function

. Identify the intervals on which the functions are concave up

and concave down.OR

2. (a) Find the limit of the rational function as and as

.

(b) Find constants a and b so that .

(c) If and , where C is the cost of producing

x units of a particular commodity and selling price p when x units are produced, determine the level of production that maximizes profit.

3. (a) If , show that .

(b) Find the amount of medicine to which the body is most sensitive by finding

the value of M that maximizes the derivative dR/dM, where

and C is a constant.(c) State and prove Snell’s law.

OR4. (a) Use L’Hopital’s rule to determine all horizontal asymptotes to the graph of

.

(b) Discuss how the graph of will look like.

(c) Use Leibniz rule to find nth derivative of .

1. The formula , where F ≥ 459.67, expresses the

Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

2. (i) Prove:

(ii) If , find the values of the other hyperbolic functions at x.

3. The position of a particle is given by the equation:

, where t is measured in seconds and s in

meters. (i) Find the acceleration at time t. What is the acceleration after 4s?(ii) When is the particle speeding up? When is it slowing down?

4. Sketch a possible graph of a function f that satisfies the following conditions:(i) on , on

(ii) on and , on

(iii) .

5. Sketch the curve: .

9.

SECTION 2

(i) Vectors in Space

1.For the given vectors, u = (4, –3, 1), v = (–2, 5. 3), find: a. u + v b. u – v c.

u d. 2u + 3v.

2.Find the standard form equation of the sphere with the given center C(0, 4, –5) and radius r = 3.

3.If u = 2i – j + 3k, v = i + j – 5k, w = 5i + 7k, find u + v – 2w.4.Sketch the cylindrical surface given by . 5.Let v = i – 2j + 2k and w = 2i + 4j – k; find ||v – w|| (v + w). Is it a vector or a scalar? 6.Determine whether (2, 3, 2), (-1, 4, 0), (-4, 5, -2) are collinear. 7. Discuss how to find a dot product, and describe an application of dot product. 8. Let v = 3i – 2j + k and w = i +j – k. Evaluate the expression (v + w).(v – w). 9. Find the scalar and vector projections of v onto w for v = i + 2k; w = -3j. 10. Find the direction cosines and the direction angles for the vector, v = 2i – 3j – 5k. 11. Fred and his friend Sam are pulling a heavy log along flat horizontal ground by

ropes attached to the front of the log. The ropes are 8ft long. Fred holds his end 2ft above the log and 1 ft to the side, and Sam holds his end 1 ft above the log and 1 ft to the opposite side. If Fred exerts a force of 30 lb and Sam exerts a force of 20 lb, what is the resultant force on the log?

12. a. Show that (v + w) · (v + w) = ||v||2 + ||w||2 + 2(v · w). b. Use part a to prove the triangle inequality: ||v + w|| ≤ ||v|| + ||w||.

13. Find v × w for the vectors v = -j + 4k; w = 5i + 6k. 14. Find a unit vector that is orthogonal to both v and w in v = 2i – 2j + k; w = 4i + 2j

– 3k. 15. Determine whether each product is a scalar or a vector or does not exist. Explain

your reasoning.a. u × (v × w)b. u · (v · w)

16. Find a number t that guarantees the vectors i + j, 2i – j + k and i + j + tk will all be coplanar.

17. Using the properties of determinants, show that u · (v × w) = (u × v) · w for any vectors u, v and w.

18. Show the validity of the equation: u × v = (u · v × i)i + (u · v × j)j + (u · v × k)k. 19. Find an explicit relationship between x and y by eliminating the parameter in x =

t4, y = t2, -1 ≤ t ≤ √2. Sketch the path described by the parametric equations over the interval.

20. Find the parametric and symmetric equations for the line passing through (1, 0, -1) and parallel to 3i + 4j.

21. Find the points of intersection of with each of the coordinate

planes.

22. Find two unit vectors parallel to the line .

23. Show that the vector v = 7i + 4j + 3k is orthogonal to the line passing through the points P(-2, 2, 7) and Q(3, -3, 2).

24. Write the given equation of plane: 4(x + 1) – 2(y + 1) + 6(z – 2) = 0 in the standard form.

25. Find two unit vectors perpendicular to the plane 2x + 4y – 3z = 4. 26. Find the distance from the point (–1, 2, 1) to the plane through the point (–3, 5, 1)

with normal vector 3i + j + 5k.

27. Find the distance between the lines and the line passing

through (1, 3, –2) and (0, 1, –1). 28. Find an equation for the line of intersection of the planes 2x – y + z = 8 and x + y

– z = 5.

29. The angle between two planes is defined to be the acute angle between their normal vectors. Find the angle between the planes 2x + y – 4z = 3 and x – y + z = 2, rounded to the nearest degree.

30. What is the distance between two points in ?31. What is the right-hand rule for a coordinate system?32. What is the formula for the distance from a point to a line? 33. Find the distance from P(–1, 1, 4) to 2x + 5y – z = 3. 34. Find equation, in both parametric and symmetric forms, of line passing through

P(1, 4, 0) with direction numbers [2, 0, 1]. 35. If u and v are orthogonal unit vectors, show that (u × v) × u = v. What is (u × v) ×

v?36. Find the center and the radius of the sphere

. 37. Find a formula for the surface area of the tetrahedron determined by vectors u, v,

w. Assume the vectors do not all lie in the same plane. 38. Find an equation for the plane that passes through the origin and whose normal

vector is parallel to the line of intersection of the planes 2x – y +z = 4 and x + 3y – z =2.

39. The vectors u, v and w are said to be linearly independent in if the only solution to the equation au + bv + cw = 0 is a = b = c = 0. Otherwise the vectors are linearly dependent. Determine whether the vectors u = –i + 2k, v = 2i – j + 3k, w = i + 3j – 2k are linearly independent or dependent.

40. A, B are vectors forming consecutive sides of a parallelogram. Find the vectors forming the other two sides.

41. A, B, C are vectors extending from an origin O. If three numbers x, y, z (not all zero) can be found such that xA + yB + zC = 0, x + y + z = 0, show that the ends of A, B, C lie in a line.

42. Prove that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the square of its four sides.

43. If u1 and u2 are vectors of unit length and θ is the angle between them, show that

.

44. Show that .

45. A particle of mass m is attracted towards the origin with a force times the distance. If it starts from R = R0 with velocity V0, find its position at time t.

46. Find the coordinates of the eight corners of the box.

47. Find the area of the triangle with vertices P(-2, 4, 5), Q(0, 7, -4) and R(-1, 5, 0). 48. Find the equation of a plane through (-1, 2, -5) and that is perpendicular to the

planes and .

49. Find the point P that lies of the distance from a point A(-1, 3, 9) to the midpoint

of the line segment joining B(-2, 3, 7) and C(4, 1, -3). 50. Find the velocity, speed and acceleration of a particle at a time t = 2 whose

position vector is . 51.

(ii) Vector-Valued Functions

1. Find the domain of the vector function F(t) = 2ti -3tj + k.

2. Describe the graph of the vector function G(t) = (sin t)i – (cos t)j. 3. Find

a. 2F(t) – 3G(t) b. G(t) · [H(t) × F(t)]

with F(t) = 2ti – 5j + t2k, G(t) = (1 – t)i + k and H(t) = (sin t)i + etj.

4. Find a vector function F whose graph is the curve of intersection of the

hemisphere and the parabolic cylinder .

5. Find: .

6. Given the vector function and ,

directly verify each of the following limit formulas. a. .

b. .

c. .

7. Find F’ and F’’ for the vector function .

8. is the position vector for a particle in space at

time . Find the particle’s velocity and acceleration vectors and then find the

speed and direction of motion for the given value of t. 9. Find the indefinite vector integral: . 10. Find the position vector R(t) and velocity vector V(t), given the acceleration

and initial position and velocity vectors

and .

11. Let v = 2i – j + 5k and w = i + 2j – 3k. Find the derivative: .

12. Let F(t) be a differentiable vector function and let h(t) be a differentiable scalar function of t. Show that [h(t)F(t)]’ = h(t)F’(t) + h’(t)F(t).

13. Find the time of flight Tf and the range Rf of a projectile fired from ground level at the angle α = with the initial speed v0 = 80ft/s. Assume that g = 32ft/s2.

14. An object moves along the given curve in the plane. Find its velocity and acceleration in terms of the unit polar vectors ur and uθ.

15. If a shot-putter throws a shot from a height of 5 ft with an angle of and initial speed of 25 ft/s, what is the horizontal distance of the throw?

16. Consider a particle moving on a circular path of radius a described by the

equation is the constant angular

velocity. Find the acceleration vector and show that its direction is always toward the centre of the circle.

17. Find two angles of elevation α1 and α2 so that a shell fired at ground level with muzzle speed of 650 ft/s will hit a target 6000 ft away.

18. is the position vector of a moving object. Find the tangential and normal components of the object’s acceleration.

19. The speed of a moving object is given. Find AT, the tangential

component of acceleration at time . 20. A boy holds onto a pail of water weighing 2 lb and swings it in a vertical circle

with radius of 3 ft. if the pail travels at ω rpm, what is the force of the water on the bottom of the pail at the highest and lowest points of the swing? What is the smallest value of ω required to keep the water from spilling from the pail? Assume the pail is held by a handle so that its bottom is straight up when it is at its highest point.

21. Define the integral of a vector function.22. Sketch the graph of , and find the length of

this curve from t = 0 to t = 2π.

23. Find the vector limit: .

24. Sketch the graph of the vector function: .

25. Evaluate: .

26. Find the tangential and normal components of acceleration, and the curvature of a moving object with position vector .

27. The position of an object moving in space is given by

. Find its velocity, acceleration, speed and direction of motion when t = 0.

28. Find F’(x) for .

29. Solve: .

30. At what angle should a projectile be fired from a ground level if its muzzle speed is 167.1 ft/s and the desired range is 600 ft.

1. The velocity of a particle moving in space is .

Find the particle’s position as a function of t if when t = 0. 2. Prove:

a.

b.3. Sketch the curve whose vector equation is . 4. Find:

a. The derivative of . b. The unit tangent vector at the point where t = 0.

5. To open the 1992 Summer Olympics in Barcelona, bronze medalist archer Antonio Rebollo lit the Olympic torch with flaming arrow. Suppose that Rebollo wanted the arrow to reach its maximum height exactly 4 ft above the center of the cauldron. a. If he shot the arrow at a height of 6 ft above ground level, express

ymax in terms of the initial speed v0 and firing angle α . b. If ymax = 74 ft, use the results of part (a) to find the value of v0 sin

α.

c. When the arrow reaches ymax, the horizontal distance travelled to the center of the cauldron is x = 90 ft. find the value of v0 cos α.

d. Find the initial firing angle of the arrow and hence find the initial speed.

5. (a) At what angle should a projectile be fired from a ground level if its muzzle speed is 167.1 ft/s and the desired range is 600 ft.

(b) Solve: .

(c) The position of an object moving in space is given by

. Find its velocity, acceleration, speed and direction of motion when t = 0.

OR6. (a) Find the tangential and normal components of acceleration, and the curvature

of a moving object with position vector

.

(b) Sketch the graph of the vector function: .

(c) Find the vector limit: .

7. (a) Find the equation of a plane through (-1, 2, -5) and that is perpendicular to the

planes and .

(b) Find equation, in both parametric and symmetric forms, of line passing through P(1, 4, 0) with direction numbers [2, 0, 1].

(c) Find the distance between the lines and the line passing

through (1, 3, –2) and (0, 1, –1).OR

8. (a) Find the point P that lies of the distance from a point A(-1, 3, 9) to the

midpoint of the line segment joining B(-2, 3, 7) and C(4, 1, -3).(b) Define scalar triple product. What is its volume interpretation? Justify. (c) Find parametric equations for the line that contains the point (3,1,4) and is parallel to the vector . Find where this line passes through the coordinate planes.

31.

Section – 3

1. (a) Let . Find .

(b) Find , assuming it exists.

(c) Let and let be a unit vector perpendicular to bothand . Find the directional derivative of f at P0 (1, -1, 2)

in the direction of .OR

2. (a) Use an incremental approximation to estimate at

. Check by using the calculator.

(b) Let be a differentiable function of x and y, and let

for and . Show that: .

(c) Determine for .9. (a) Classify each point of as a relative maximum, a

relative minimum, or a saddle point. (b) Compute the slope of the tangent line to the graph of at the point P0 (1,-1, -2) in the direction plane parallel to the xz-plane. (c) State and prove chain rule for one independent parameter.

OR10. (a) Describe the level curves of the given quadratic surface in

each coordinate plane.

(b) Given that the function is continuous at the

origin, what is B?(c) State and prove the normal property of the gradient.