I Plot the function y=10 x^6 for x=0 to 3 ii Plot on...

i Plot the function $y=10 x^{6}$ for $x=0$ to 3 ii Plot on different axes the function $y=10 x^{6}$ for $x=1.4$ to $1.6$ $x=1.49$ to $1.51$ and $x=1.499$ to $1.501$. iii Evaluate $\lim _{h \rightarrow 0} \frac{10(1.5+h)^{6}-10(1.5)^{6}}{h}$. This is $\frac{\mathrm{d} y}{\mathrm{~d} x}$ of $y=10 x^{6}$ at $x=1.5$.

i Plot the function $y=10 x^{6}$ for $x=0$ to 3
ii Plot on different axes the function $y=10 x^{6}$ for $x=1.4$ to $1.6$ $x=1.49$ to $1.51$ and $x=1.499$ to $1.501$.
iii Evaluate $\lim _{h \rightarrow 0} \frac{10(1.5+h)^{6}-10(1.5)^{6}}{h}$.
This is $\frac{\mathrm{d} y}{\mathrm{~d} x}$ of $y=10 x^{6}$ at $x=1.5$.

Key Concepts

Differentiation Using the Power Rule

The power rule is a basic technique in differential calculus that simplifies the process of differentiating functions of the form x^n. It states that the derivative of x raised to a power is n times x raised to the (n-1) power. This rule is instrumental when applying the limit definition of the derivative, as it allows for quick computation of derivatives for polynomial functions.

Limit Definition of the Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the incremental change approaches zero. This concept, known as the limit definition of the derivative, forms the foundation of differential calculus and provides an exact measure of the instantaneous rate of change.

Plotting Functions

This concept involves graphically representing a mathematical function on coordinate axes. It helps visualize how the dependent variable changes with the independent variable over a specified interval. In teaching and analysis, plotting functions over different intervals, including wider domains and very narrow regions, provides insight into overall behavior and local details of the function.

Zooming Into Function Behavior

Zooming into a specific region of a function’s graph allows for a closer examination of its properties, such as subtle variations, trends, or changes in curvature that may not be obvious in a full-scale view. This is especially beneficial for identifying rapid changes or confirming smoothness of the function at or near points of interest.

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