example 2 sketch the graph of each of the following vector functions.\n(b) $\\vec{r}(t) = \\langle t, t^3…

Explanation:

Step1: Identify parametric equations

The vector function $\vec{r}(t) = \langle t, t^3 - 10t + 7 \rangle$ gives $x = t$ and $y = t^3 - 10t + 7$.

Step2: Eliminate the parameter

Substitute $t = x$ into the equation for $y$ to get the Cartesian form. $$y = x^3 - 10x + 7$$

Step3: Find critical points

Differentiate $y$ with respect to $x$ and set to zero. $$\frac{dy}{dx} = 3x^2 - 10 = 0 \implies x = \pm \sqrt{\frac{10}{3}} \approx \pm 1.83$$

Step4: Determine local extrema

Evaluate $y$ at critical points: $y(1.83) \approx -5.17$ (min) and $y(-1.83) \approx 19.17$ (max).

Step5: Identify intercepts

The $y$-intercept is at $(0, 7)$. The graph behaves like $y = x^3$ for large $|x|$.

Answer:

The graph is a cubic curve defined by $y = x^3 - 10x + 7$. It has a local maximum at approximately $(-1.83, 19.17)$, a local minimum at approximately $(1.83, -5.17)$, and passes through the $y$-axis at $(0, 7)$. The curve enters from the bottom-left (third quadrant) and exits toward the top-right (first quadrant).