91Ó°ÊÓ

Exponential functions represent rapid growth or decay and are described by the equation of the form (1)
Exponential functions like the one given in the exercise, (2)
are essential in modeling diverse phenomena, such as population growth, radioactive decay, and interest rates. The function (3)
has a base of 3, which indicates the function will grow exponentially as (4)
increases. Its graph is ever-increasing and will asymptotically approach the x-axis as (5)
becomes very negative, without ever actually reaching it—this line is called a horizontal asymptote. To sketch, start by plotting a few points for integer values of (6)
and drawing a smooth curve that gets steeper the further right you go (while never touching the x-axis on the left).

Derivatives represent the rate of change of a function with respect to its independent variable. In calculus, when you take the derivative of a function, you're calculating the slope of the tangent line at any point on its graph. For an exponential function like (1)
the derivative (2)
represents how quickly the function values are changing at any given time
Finding the derivative (3)
at a specific point, particularly where
is 1 or 0 as in the exercise, gives us information on the instantaneous growth rate at that point. To obtain a numerical approximation of (4)
the exercise suggests calculating the slope of the secant as an estimation, which is the average rate of change over a tiny interval around the point of interest. This introduces the concept of limits, which is central to the definition of a derivative.

The slope of a line measures its steepness, usually denoted by the letter
In the context of the given exercise, the slope is critical for understanding two concepts: the slope of the secant and the slope of the tangent. The slope of the secant line—the line connecting two points on the curve—gives an average rate of change between those points. It is calculated with the formula: (1)
where
and
are intervals on the independent variable and
is typically a small number. The slope of the tangent, on the other hand, measures the instantaneous rate of change at a specific point and is found using the derivative. The approximation technique used in the exercise computes the slope of the secant with an exceedingly small
essentially providing a close estimate to the actual slope of the tangent at a specific point. Considering the difference between these slopes is key in understanding the concept of differentiation.

Graph sketching is a vital skill in mathematics as it provides a visual representation of functions and their properties. Through sketching, one can quickly infer behavior such as increasing or decreasing intervals, asymptotes, and loci of points of interest like maxima, minima, or inflection points. For the function (1)
sketch the graph starting with its basic shape—an increasing curve that gets steeper as
increases. Then mark important points like
and use your understanding of exponential growth to shape the curve appropriately. When sketching the slope function or the derivative (2)
the basic principle is that if the function is increasing, its derivative should be above the
axis (indicating a positive rate of change), and if decreasing, then below. Since an exponential function always increases, the slope function will always be above the
axis and increase as well. By linking the derivative's graphical traits to the original function's behavior, graph sketching becomes a powerful tool in analyses and understanding function characteristics.