To determine
To write: Expression for the linearization of f at a.
Answer
Lx=fa+f'(a)(x-a)
Explanation
Close to the point of tangency a curve is nearby to its tangent line. In other words, the graph of a differentiable function looks like its tangent line near the point of tangency. From this we can find the approximate values of functions.
We can calculate a value f(a) of a function, but it might be hard to find the nearby values of f. So we consider the tangent which is a linear function. It is easy to calculate it’s values.
In other words, use tangent line at (a,fa) as an approximation to the curve y=f(x) when x is near a. An equation of tangent line is
y=fa+f'(a)(x-a)
and the approximation
f(x)≈fa+f'(a)(x-a)
called linear approximation or tangent approximation of f at a.The linear function whose graph is tangent line at x=a, that is,
Lx=fa+f'(a)(x-a)
is called linearization of f at a.
Conclusion:
Expression for the linearization of f at a is Lx=fa+f'(a)(x-a)
To determine
To write: Expression for differential dy.
Answer
dy=f'xdx
Explanation
Consider y=f(x). Here f is a differentiable function. Hence dx becomes an independent variable and it can be given the value of any real number.
The differential dy is then defined as:
dy=f'x dx
Here,dy is a dependent variable. It will depend on the values of x and dx. Conclusion:
Expression for differential dy is dy=f'x dx
To determine
To write: Draw a picture showing the geometrical meaning of ∆y and dy.
Answer
dy ≈∆y
Explanation
Given:dx=∆x
The, geometric meaning of differentials is shown above.
Let P(x,f(x)) and Q(x+∆x,fx+∆x) be points on the graph of f and let dx=∆x. The corresponding change in y is
∆y=fx+∆x-f(x)
The derivative f’(x) is the slope of the tangent line PR. Thus the directed distance from S to R is f'x=dy.
The value of the tangent line which rises or falls is given by dy, when x changes by an amount dx, whereas ∆y represents the amount that the curve y=f(x) rises or falls when x changes by an amount dx=∆x.
Conclusion:
If dx=∆x then dy ≈∆y.