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

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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 - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Thus Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics(i)
From the equation of curve Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics (ii)
From equation (i) and (ii)
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Thus there are two points Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study 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 - Notes | Study 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 - Notes | Study Mathematical Models - Physics

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

Ans. Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Hence, 4a • 0+3b • 0+c = 0 ⇒ c = 0
And, 4a (-1) + 3b = 0    (i)
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study 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 - Notes | Study 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 - Notes | Study Mathematical Models - Physics is defined for all x , hence for local maximum or minimum value Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - PhysicsWhere we have used the fact that.
Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics
Thus Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study Mathematical Models - Physics
Thus at Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physicsa local minimum occurs. The local minimum value is Calculus of Single & Multiple Variables -Assignment - Notes | Study 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 - Notes | Study Mathematical Models - Physics

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

The document Calculus of Single & Multiple Variables -Assignment - Notes | Study Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
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