Question

Suppose f(x)=a and f¹(x)=b, estimate f(x+h).

A) f(x+h)=a+bh
B) f(x+h)=a+b
C)f(x+h)=ah+b
D) f(x+h)=abh

Answer 1

To estimate f(x+h), we use linear approximation which leads to the formula f(x+h) = f(x) + f'(x)(h), simplifying to f(x+h) = a + bh, corresponding to option A.

Step-by-step explanation:

To estimate f(x+h) given that f(x)=a and f'(x)=b, we can use the linear approximation, which is essentially the equation of the tangent line at x. The tangent line's equation at the point (x, f(x)) with slope f'(x) is f(x) + f'(x)(h-x). Since we are estimating f(x+h), we set h as the difference between x+h and x. Simplifying the equation, we get:

f(x+h) = f(x) + f'(x)(h-x)

But since x is our initial point, h-x simplifies to just h. So we have:

f(x+h) = a + bh

This corresponds to option A) f(x+h) = a + bh.