Mathematics

Problem

The notion of the derivative is often introduced through the idea of slope. Answer the following questions to demonstrate your understanding of this connection. a. (i) Define the derivative of a real-valued function

f

at a point

x=a

by means of a limit. b. (ii) Explain, in words, how this limit represents the slope of the tangent to the curve

y=f(x)

at the point

(a,f(a))

. c. (iii) For the specific function

f(x)=x^{2}

, consider the secant line joining the points

(1,1)

and

(1+h,(1+h)^{2})

. Describe what happens to the slope of this secant line as

h\to0

and relate your observation to the slope of the tangent at

x=1

(no numerical calculation of the limit is required).

Intuitive explanation

The derivative captures how a function's output changes relative to its input—essentially the slope of the curve at any single point. Rather than measuring slope between two distant points, you're zooming in infinitesimally close to see the instantaneous rate of change. Understanding this limit process is the key: as two points on the curve get arbitrarily close together, their secant line becomes the tangent line, and that's your derivative.

Intuitive explanation

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