Question

1. graphical reasoning use a graphing utility to graph each function and its tangent lines at x = - 1, x = 0, and x = 1. based on the results, determine whether the slopes of tangent lines to the graph of a function at different values of x are always distinct. (a) f(x)=x^2 (b) g(x)=x^3

Explanation:

Step1: Recall the derivative formula

The derivative of a function $y = f(x)$ gives the slope of the tangent - line at a point $x$. The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$.

Step2: Find the derivative of $f(x)=x^{2}$

Using the power - rule, if $f(x)=x^{2}$, then $f^\prime(x) = 2x$.
Evaluate $f^\prime(x)$ at $x=-1,0,1$:
When $x = - 1$, $f^\prime(-1)=2\times(-1)=-2$.
When $x = 0$, $f^\prime(0)=2\times0 = 0$.
When $x = 1$, $f^\prime(1)=2\times1 = 2$.

Step3: Find the derivative of $g(x)=x^{3}$

Using the power - rule, if $g(x)=x^{3}$, then $g^\prime(x)=3x^{2}$.
Evaluate $g^\prime(x)$ at $x=-1,0,1$:
When $x=-1$, $g^\prime(-1)=3\times(-1)^{2}=3$.
When $x = 0$, $g^\prime(0)=3\times0^{2}=0$.
When $x = 1$, $g^\prime(1)=3\times1^{2}=3$.

Step4: Analyze the slopes of the tangent lines

For $f(x)=x^{2}$, the slopes of the tangent lines at $x=-1,0,1$ are $-2,0,2$ respectively, which are distinct.
For $g(x)=x^{3}$, the slopes of the tangent lines at $x=-1$ and $x = 1$ are both $3$, so the slopes of the tangent lines to the graph of $g(x)$ are not always distinct.

Answer:

For $f(x)=x^{2}$, the slopes of the tangent lines at $x=-1,0,1$ are distinct. For $g(x)=x^{3}$, the slopes of the tangent lines at $x=-1,0,1$ are not always distinct.