Rates of Growth (a) By drawing the graphs of the functions...

Rates of Growth (a) By drawing the graphs of the functions $$ f(x)=1+\ln (1+x) \quad \text { and } \quad g(x)=\sqrt{x} $$ in a suitable viewing rectangle, show that even when a logarithmic function starts out higher than a root function, it is ultimately overtaken by the root function. (b) Find, rounded to two decimal places, the solutions of the equation $\sqrt{x}=1+\ln (1+x)$

Rates of Growth
(a) By drawing the graphs of the functions
$$
f(x)=1+\ln (1+x) \quad \text { and } \quad g(x)=\sqrt{x}
$$
in a suitable viewing rectangle, show that even when a logarithmic function starts out higher than a root function, it is ultimately overtaken by the root function.
(b) Find, rounded to two decimal places, the solutions of the equation $\sqrt{x}=1+\ln (1+x)$

Key Concepts

Numerical Methods

When equations involving transcendental functions, such as logarithms and roots, do not have simple algebraic solutions, numerical methods are used to approximate the solutions. This often involves iterative techniques or computational approaches to find accurate approximations, which can then be rounded to a specified number of decimal places.

Rate of Growth

This concept refers to comparing the speed at which functions increase as the input becomes very large. In the context of logarithmic and root functions, it emphasizes that even if a logarithmic function starts out with higher values, its slow growth rate means that eventually the steadily rising root function will overtake it.

Logarithmic Functions

Logarithmic functions are defined as the inverse of exponential functions and exhibit a very slow rate of growth when compared to power functions. Their significance lies in their unique scaling properties, which means that despite possibly high initial values, their incremental increases become smaller as the input increases.

Root Functions

Root functions, such as the square root function, are examples of power functions that increase at a moderate rate. They are characterized by an exponent between 0 and 1 and demonstrate that even though their growth may seem gradual, in the long run they can surpass functions like logarithms due to their steadily increasing nature.

Graphical Analysis

Graphical analysis involves plotting functions on a coordinate system to visually inspect and compare their behaviors. This method is particularly useful for identifying points of intersection and understanding the relative growth of functions over given domains.

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