Sketch the graph of ∫by hand and use your sketch to find the...

Sketch the graph of $\int$ by hand and use your sketch to
find the absolute and local maximum and minimum values of $f .$
(Use the graphs and transformations of Sections 1.2 and $1.3 . )$
$$f(t)=\cos t, \quad-3 \pi / 2 \leqslant t \leqslant 3 \pi / 2$$

Key Concepts

Graphical Analysis

Graphical analysis involves interpreting the shape and key features of a function's graph, such as regions where the function is increasing or decreasing, as well as identifying local and absolute extrema. This approach is essential in understanding function behavior through visual means.

Integration as an Accumulation Function

Viewing an integral as an accumulation function allows one to interpret it as the net area under a given curve up to a certain point. This concept bridges the process of integration with the notion of area, enabling the sketching of functions represented by integrals.

Trigonometric Functions

Trigonometric functions are characterized by their periodic and oscillatory nature, with well-defined amplitudes and periods. Understanding these properties is important when sketching their graphs and analyzing their extreme values over specific intervals.

Finding Extrema

The process of finding absolute and local maximum and minimum values of a function involves examining critical points, endpoints of the interval, and regions where the function's derivative changes sign. This technique is fundamental in optimization problems and in understanding the overall behavior of functions.

Transformation of Graphs

Transformations such as translations, reflections, and scalings are used to modify and sketch basic function graphs. Recognizing these transformations can simplify the process of graphing more complex functions by relating them to simpler, well-known functions.

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