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

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. find the following in calculating the area of the circular plate:
Absolute error

The radius of a circular plate is measured as 12.65 cm instead of the actual length 12.5 cm. Find the following in calculating the area of the circular plate:
Relative error

Find differential dy for each of the following function \(y=\frac { { \left( 1-2x \right) }^{ 3 } }{ 3-4x } \)

Find differential dy for each of the following function
y = (3 + sin(2x)) ^2/3 

Find differential dy for each of the following function
y = _ex^2_-5x+7 cos (x² - 1)

The relation between the number of words y a person learns in x hours is given by y = 52 \(\sqrt { x } \), 0, ≤ x ≤ 9. What is the approximate number of words learned when x changes from
(i) 1 to 1.1 hour?
(ii) 4 to 4.1 hour?

The relation between the number of words y a person learns in x hours is given by y = 52 \(\sqrt { x } \), 0, ≤ x ≤ 9. What is the approximate number of words learned when x changes from

Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = 0 along every line y = mx, m ∈ R

Let g(x, y) = \(\frac { { x }^{ 2 }y }{ { x }^{ 4 }+{ y }^{ 2 } } \) for (x, y) ≠ (0, 0) and f(0, 0) = 0
Show that \(\begin{matrix} lim \\ (x,y)\rightarrow (0,0) \end{matrix}\) g(x, y) = \(\frac { k }{ 1+{ k }^{ 2 } } \) along every parabola y = kx², k ∈ R \ {0}.

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)

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.