Find the linear approximation for f(x) = \(\sqrt { 1+x } ,x\ge -1\) at x₀ = 3. Use the linear approximation to estimate f(3.2) 

Use linear approximation to find an approximate value of \(\sqrt { 9.2 } \) without using a calculator.

Let us assume that the shape of a soap bubble is a sphere. Use linear approximation to approximate the increase in the surface area of a soap bubble as its radius increases from 5 cm to 5.2 cm. Also, calculate the percentage error.

Let \(f(x)=\sqrt [ 3 ]{ x } \). Find the linear approximation at x = 27. Use the linear approximation to approximate \(\sqrt [ 3 ]{ 27.2 } \)

Find a linear approximation for the following functions at the indicated points.
f(x) = x³ - 5x + 12, x₀ = 2

Find a linear approximation for the following functions at the indicated points.
g(x) = \(g(x)=\sqrt { { x }^{ 2 }+9 } ,{ x }_{ 0 }=-4\)

Find a linear approximation for the following functions at the indicated points.
\(h(x)=\frac{x}{x+1}, x_{0}=1\)

Let f, g : (a, b)→R be differentiable functions. Show that d(fg) = fdg + gdf

If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

Find df for f(x) = x² + 3x and evaluate it for
x = 2 and dx = 0.1

Find df for f(x) = x² + 3x and evaluate it for
x = 3 and dx = 0.02

Find ∆f and df for the function f for the indicated values of x, ∆x and compare
(1) f(x) = x³ - 2x² ; x = 2, ∆ x = dx = 0.5
(2) f(x) = x² + 2x + 3; x = -0.5, ∆x = dx = 0.1 

Find ∆f and df for the function f for the indicated values of x, ∆x and compare

Assuming log₁₀e = 0.4343, find an approximate value of log₁₀ 1003

An egg of a particular bird is very nearly spherical. If the radius to the inside of the shell is 5 mm and radius to the outside of the shell is 5.3 mm, find the volume of the shell approximately.

Assume that the cross section of the artery of human is circular. A drug is given to a patient to dilate his arteries. If the radius of an artery is increased from 2 mm to 2.1 mm, how much is cross-sectional area increased approximately?

Let f (x,y) = \(\frac { 3x-5y+8 }{ { x }^{ 2 }+{ y }^{ 2 }+1 } \) for all (x, y) ∈ R² Show that f is continuous on R² 

f(x,y) = \(\frac { xy }{ { x }^{ 2 }+{ y }^{ 2 } } \), if (x,y) ≠ (0, 0) and f (0, 0) = 0. Show that f is not continuous at (0, 0) and continuous at all other points of R²

Evaluate \(\begin{gathered} \text { lim } \\ (x, y) \rightarrow(1,2) \end{gathered}\)g(x, y), if the limit exists, where g\((x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 } \)

Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}cos=\left( \frac { { x }^{ 3 }+{ y }^{ 3 } }{ x+y+2 } \right) \). If the limit exists.

Show that f(x, y) = \(\frac { { x }^{ 2 }-{ y }^{ 2 } }{ { y }^{ 2 }+1 } \) is continuous at every (x, y) ∈ R²

Let g(x, y) = \(\frac { { e }^{ y }sinx }{ x } \), for x ≠ 0 and g(0, 0) = 1. Show that g is continuous at (0, 0).

For each of the following functions find the f_x, f_y, and show that f_xy = f_yx
f(x, y) = \(\frac { 3x }{ y+sinx \ } \) 

For each of the following functions find the f_x, f_y, and show that f_xy = f_yx
f(x, y) = tan ^-1 (x/y) 

For each of the following functions find the f_x, f_y and show that f_xy = f_yx
f(x, y) = cos (x² - 3xy)

If U(x, y, z) = \(\frac { { x }^{ 2 }+{ y }^{ 2 } }{ xy } +3{ z }^{ 2 }y\), find \(\frac { \partial U }{ \partial x } ;\frac { \partial U }{ \partial y } \) and \(\frac { \partial U }{ \partial z } \)

For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = xe^y + 3x²y

For each of the following functions find the g_xy, g_xx, g_yy and g_yx.
g(x, y) = log (5x + 3y)

For each of the following functions find the g_xy, g_xx, g_yy and g_yx. 
g(x, y) = x² + 3xy − 7y + cos(5x)

A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x² − 0.05y² and C(x, y) = 8x + 6y + 2000 respectively

Let U(x, y, z) = x² − xy + 3 sin z, x, y, z ∈ R Find the linear approximation for U at (2,−1,0).

If w(x, y) = x³ − 3xy + 2y², x, y ∊ R, find the linear approximation for w at (1,−1)

Let z(x, y) = x²y + 3xy⁴, x, y ∈ R. Find the linear approximation for z at (2, -1).

Let V(x, y, z) = xy+ yz + zx, x, y, z ∈ R. Find the differential dV.

Verify the above theorem for F(x, y) = x² - 2y² + 2xy and x(t) = cos t, y(t) = sin t, t ∈ [0, 2\(\pi\)]

Let g( x, y) = x² - yx + sin(x+y), x(t) = e^3t, y(t) = t², t ∈ R. Find \(\frac { dg }{ dt } \) 

Let g(x, y) = 2y + x², x = 2r -s, y = r²+ 2s, r, s ∊ R. Find \(\frac { \partial g }{ \partial r } ,\frac { \partial g }{ \partial s } \)

Let U(x, y, z) = xyz, x = e^-t, y = e^-t cos t, z = sin t, t ∈ R. Find \(\frac{dU}{dt}\)

If z(x, y) = x tan-¹ (xy), x = t², y = se^t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \) at s = t = 1

Let U(x, y) = e^x sin y, where x = st², y = s² t, s, t ∈ R. Find \(\frac { \partial U }{ \partial s } ,\frac { \partial U }{ \partial t } \) and evaluate them at s = t = 1.

Let z(x, y) = x³ - 3x²y³, where x = se^t, y = se^-t, s, t ∈ R. Find \(\frac { \partial z }{ \partial s } \) and \(\frac { \partial z }{ \partial t } \)

Prove that f(x, y) = x³ - 2x²y + 3xy² + y³ is homogeneous; what is the degree? Verify Euler's Theorem for f.

prove that g(x, y) = x log\(\left( \frac { y }{ x } \right) \) is homogeneous; what is the degree? Verify Euler's Theorem for g.

If v(x, y) = log \(\left( \frac { { x }^{ 2 }+{ y }^{ 2 } }{ x+y } \right) \), prove that \(x\frac { \partial v }{ \partial x } +y\frac { \partial u }{ \partial y } \) = 1

If w(x,y, z) = log \(\left( \frac { { 5x }^{ 3 }{ y }^{ 4 }+7{ y }^{ 2 }{ xz }^{ 4 }-{ 75y }^{ 3 }{ z }^{ 4 } }{ { x }^{ 2 }+{ y }^{ 2 } } \right) \) find \(x\frac { \partial w }{ \partial x } +y\frac { \partial w }{ \partial y } +z\frac { \partial w }{ \partial z } \)

If u = sin^-1 \(\left( \frac { x+y }{ \sqrt { x } +\sqrt { y } } \right) \), Show that  \(x\frac { \partial u }{ \partial x } +y\frac { \partial u }{ \partial y } =\frac { 1 }{ 2 } tanu\)

State Function of Function Rule Theorem.