Question

The following figure shows the graphs of \[{\rm{f,f'}}\] and \[\int_{\rm{0}}^{\rm{x}} {{\rm{f(t)dt}}} \]. Identify each graph, and explain your choices.

Short answer

Identification of each graph is –

Function\[{\rm{f(x)}} \to {\rm{c}}\]

Function \[{\rm{f'(x)}} \to {\rm{b}}\]

Function \[\int_{\rm{0}}^{\rm{x}} {{\rm{f(t)dt}}} \to {\rm{a}}\]

Step-by-step solution

Concept Introduction

In mathematics, an integral is a numerical number equal to the area under a function's graph for some interval or a new function whose derivative equals the original function.

Identifying graphs of each function

Evaluate and study the graph –

Explanation for each graph

At local minima/maxima of every function their derivative is zero or there is direction change.

Here function \[{\rm{f}}\] is denoted by the curve \[{\rm{c}}\]and it can be seen that when graph \[{\rm{c}}\] has local maxima and minima, black graph has value zero (changing direction).

So, \[{\rm{f(x)}}\] is denoted by curve c light blue curve and \[{\rm{f'(x)}}\] is black curve \[{\rm{b}}\]. We know that \[\frac{{\rm{d}}}{{{\rm{dx}}}}\int_{\rm{a}}^{\rm{x}} {{\rm{f(t)dt = f(x)m}}} \].

So, when the curve of \[\int_{\rm{a}}^{\rm{x}} {{\rm{f(x)dx}}} \] has local maximum and minimum function \[{\rm{f(x)}}\] will be zero.

From the graph it is clear, when curve \[{\rm{a}}\] has maxima or minima the curve c function f \[{\rm{f(x)}}\]has direction change.

Therefore, the curves that represent each function are \[{\rm{f(x)}} \to {\rm{c}}\], \[{\rm{f'(x)}} \to {\rm{b}}\] and \[\int_{\rm{0}}^{\rm{x}} {{\rm{f(t)dt}}} \to {\rm{a}}\].