Physics Exam  >  Physics Notes  >  Mathematical Models  >  Calculus of Single & Multiple Variables -Assignment

Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics PDF Download

Q.1. If f (x) be a twice differentiable function such that f"(x) = -f (x) and f'(x) = g (x ). If h'(x) = [f(x)]2+[g(x)]2 ’h (1) = 8 and h (0) = 2 then find the value of h (2) .
Ans. h'( X ) = [f ( x )]2+[ g ( x )]2
h'' ( x ) = 2 f (x ) f' ( x ) + 2 g ( x ) g ' (x )
h'' ( x ) = 2 f(x) g (x ) + 2g ( x ) f'' (x )
h'' ( x ) = 2 f(x) g (x ) + 2g ( x ) (-f (x )) = 0
Thus, h'(x) = c, a constant for all x.
⇒ h (x) = cx + c1
It is given that h (l) = 8 and h (0) = 2, therefore c1 = 2 and c2 = 6
∴ h (x ) = 6 x + 2 ⇒ h (2) = 6 x 2 + 2 = 14

Q.2. If y = ax then find the value of dy/dx as a function of x and y ( a is a constant).
Ans. We can write y = axy
Taking Logrithm of both sides, we obtain ln y = xy ln a.
⇒ ln (ln y) = y ln x + ln (ln a)
Differentiating with respect to x gives
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

Q.3. Find the co-ordinates of the point P on the curve y2 = 2x3 such that the tangent at P is perpendicular to the line 4 x - 3 y + 2 = 0.
Ans. Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Let, P(x0,y0) be the point at which the tangent is perpendicular to the line 4x - 3y + 2 = 0.
Thus slope at P is equal to m = Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Thus Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics(i)
From the equation of curve Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics (ii)
From equation (i) and (ii)
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Thus there are two points Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics and (0,0) at which the tangent line is perpendicular
to the line 4x - 3y + 2 = 0.

Q.4. If the tangent to the curve xy + ax + by = 0 at (1,1) is inclined at an angle tan-1 2 with x axis then find the values of a and b .
Ans. The point (1,1) lies on the curve xy + ax + by = 0.
Hence, a+b = -1    (i)
Differentiating xy + ax + by = 0 with respect to x we get
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Since the tangent at (1,1) makes an angle of tan-1 2 with the x - axis hence
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Solving equations (i) and (ii) we obtain
a = 1 and b = -2

Q.5. If f (x) = a log |x| + bx2 + x has its extreme values (local maximum or minimum value) at x = -1 and x = 2 , then find the values of a and b .
Ans. The logarithmic function is defined for all x > 0. Hence the domain of f (x) is (0, ∞) . Using this fact we can write
f (x) = a log x + bx2 + x
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
We see that the derivative is defined at all points of the domain of the function. Hence the function can attain extreme values only at points where f'(x) = 0. Since f (x) attains its extreme values at x = -1,2.
∴ f'(-1) = 0 and f'(2) = 0
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

Q.6. If y = sin-1 Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics where α is a constant then find the value of y'(0) .

Ans. Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

Q.7. Find the equation of the tangent to the curve x = t cos t and y = t sin t at the origin of xy plane.
Ans. Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

At the origin x = 0, y = 0
⇒ t cos t = 0 and t sin t = 0
t = 0 and cos t = 0 and t = 0 and sin t = 0
Since there is no value of t for which cos t and sin t are simultaneously 0 , hence t = 0.Thus the slope of the tangent at (0,0) is Calculus of Single & Multiple Variables -Assignment | Mathematical Models - PhysicsThus the equation of the tangent at the origin is y = 0.

Q.8. A curve with equation of the form y = ax4 + bx3 + c + cx + d has slope 0 at the point (0,1) and also touches the x -axis at the point (-1,0) then find the values of x for which the curve has negative slope.
Ans: Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Hence, 4a • 0+3b • 0+c = 0 ⇒ c = 0
And, 4a (-1) + 3b = 0    (i)
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Also, the curve passes through (0,1) and (-1,0), hence
d = 1 and 0 = a - b - c + d
⇒ a - b - c + 1 = 0    (ii)
From (i) and (ii), we get
a = 3,b = 4,c = 0 d = 1
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

Q.9. Find the point at which the local maximum or local minimum value of the function
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
occurs. Also find the local minimum and for maximum values of the function.
Ans. We have y = f (x) = sinx + cos4 x
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
= -4 cos x sin x (cos2 x - sin2 x)
= -2 sin 2x cos 2x = - sin 4x
Over the given interval Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics is defined for all x , hence for local maximum or minimum value Calculus of Single & Multiple Variables -Assignment | Mathematical Models - PhysicsWhere we have used the fact that.
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Thus Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics is the only point where local maximum or minimum value of the function occurs. Here we can use either first derivative test or the second derivative test to check for local maxima or minimum. Using the second derivative test, we obtain
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Thus at Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physicsa local minimum occurs. The local minimum value is Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Note: Students should check using the first derivative test that a local minimum indeed occurs at x = Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

Q.10. If A > 0,B > 0 Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics then find the maximum value of tan A + tan B.
Ans. We have A + B =Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Let, z = tan A+tan B
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics
F or maximum value of Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics because Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physicswhich implies that x = tan A > 0. It can be earily checked that Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics Hence, z is maximum for x = Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics For this value of x, z = Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

The document Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
All you need of Physics at this link: Physics
18 docs|24 tests

Up next

FAQs on Calculus of Single & Multiple Variables -Assignment - Mathematical Models - Physics

1. What is calculus of single and multiple variables?
Ans. Calculus of single and multiple variables is a branch of mathematics that deals with the study of change and motion. It involves the concepts of differentiation and integration, which are used to analyze functions and their properties in one or more variables.
2. What is the difference between calculus of single and multiple variables?
Ans. The main difference between calculus of single and multiple variables lies in the number of variables involved. Single-variable calculus focuses on functions of a single variable and deals with concepts like limits, derivatives, and integrals in one dimension. On the other hand, multiple-variable calculus deals with functions of multiple variables and involves concepts like partial derivatives, multiple integrals, and vector calculus.
3. What are the applications of calculus of single and multiple variables?
Ans. The applications of calculus of single and multiple variables are extensive and diverse. In single-variable calculus, it is used to model and analyze physical phenomena such as motion, growth, and decay. It is also fundamental in the fields of physics, engineering, economics, and computer science. In multiple-variable calculus, it is applied to study functions of multiple variables, optimization problems, and vector fields, which find applications in fields like fluid mechanics, electromagnetism, and optimization algorithms.
4. What are the prerequisites for studying calculus of single and multiple variables?
Ans. To study calculus of single and multiple variables, a strong foundation in algebra, trigonometry, and basic calculus concepts is essential. It is important to have a solid understanding of functions, limits, derivatives, and integrals in one variable before diving into the multi-variable calculus. Additionally, knowledge of vectors and matrices can also be beneficial in understanding certain concepts in multiple-variable calculus.
5. How can I prepare for the IIT JAM exam in calculus of single and multiple variables?
Ans. To prepare for the IIT JAM exam in calculus of single and multiple variables, it is recommended to follow a structured study plan. Start by thoroughly understanding the fundamental concepts of single-variable calculus, including limits, derivatives, and integrals. Practice solving a variety of problems from textbooks and previous year question papers. Once comfortable with single-variable calculus, move on to studying concepts of multiple-variable calculus, such as partial derivatives, multiple integrals, and vector calculus. Regularly solve mock tests and sample papers to assess your progress and identify areas where you need more practice.
18 docs|24 tests
Download as PDF

Up next

Explore Courses for Physics exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Download the FREE EduRev App
Track your progress, build streaks, highlight & save important lessons and more!
Related Searches

Exam

,

Previous Year Questions with Solutions

,

Semester Notes

,

Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

,

pdf

,

video lectures

,

ppt

,

shortcuts and tricks

,

Viva Questions

,

Objective type Questions

,

study material

,

Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

,

Extra Questions

,

mock tests for examination

,

practice quizzes

,

Sample Paper

,

Free

,

Calculus of Single & Multiple Variables -Assignment | Mathematical Models - Physics

,

past year papers

,

MCQs

,

Summary

,

Important questions

;