Calculus – 2110014“Total Differential ,Tangent Plane, Normal Line, Linear

Approximation,”Prepared By:NiraliAkabari

Tangent planes

Tangent planes

Suppose a surface S has equation z = f(x, y), where f has continuous first partial derivatives.

Let P(x0, y0, z0) be a point on S.

Let T1 and T2 be the tangent lines to the curves C1 and C2 at the point P.

Tangent planes

Tangent planes

Tangent planes

An equation of the tangent plane to the surface z = f(x, y) at the point P(x0, y0, z0) is: fx(x0, y0, z0)(x – x0) + fy(x0, y0, z0)(y – y0) + fz (x0, y0, z0) ( z– z0 )= 0

Normal line The normal line to a curve at a particular point is

the line through that point and perpendicular to the tangent.

A person might remember from analytic geometry that the slope of any line perpendicular to a line with slope m is the negative reciprocal −1/m.

Thus, just changing this aspect of the equation for the tangent line, we can say generally that the equation of the normal line to the graph of ’f’ at (x0 ,f(x0 )) is

y − f(x0 ) = −1 (x−x0 ).

f′(x0 )

Normal line

Normal line Let f (x,y,z) define a surface that is

differentiable at a point (x0,y0,z0), then the normal line to f(x,y,z) at ( x0 , y0 , z0 ) is the line with normal vector

f (x0,y0,z0)that passes through the point (x0,y0,z0).  In Particular the equation of the normal line is

          x(t) = x0 + fx(x0,y0,z0) t  

          y(t) = y0 + fy(x0,y0,z0) t  

          z(t) = z0 + fz(x0,y0,z0) t

Normal line

Linear Approximations

Linear ApproximationsThe idea is that it might be easy to calculate a value f(a) of a function, but difficult (or even impossible) to compute nearby values of f.

So, we settle for the easily computed values of the linear function L whose graph is the tangent line of f at (a, f(a)).

In other words, we use the tangent line at (a, f(a)) as an approximation to the curve y = f(x) when x is near a.

An equation of this tangent line is y = f(a) + f’(a)(x - a)

Linearization

The linear function whose graph is this tangent line, that is, L(x) = f(a) + f’(a)(x – a) is called the linearization of f at a.

Take a look at the following graph of a function and its tangent line.

• From this graph we can see that near x=a the tangent line and the function have nearly the same graph.  On occasion we will use the tangent line, L(x) , as an approximation to the function, f(x), near x=a .

• In these cases we call the tangent line the linear approximation to the function at x=a.

 

Total Differential

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Thank you