Homework8
Name:
1. Find the derivatives of the following functions using differentiation rules.
a) $y=\frac{x^{3}}{3}-x$
b) $y=(x+1)\left(x^{2}+1\right)$
c)
$$
y=\frac{x^{2}+5 x-1}{x^{2}}
$$
2. Suppose $u$ and $v$ are functions of $x$ that are differentiable at $x=0$, and that $u(0)=5, u^{\prime}(0)=-3, v(0)=-1$, $v^{\prime}(0)=2$. Find the values of the following derivatives at $x=0$.
(a) $\frac{d}{d x}(u v)$
(b) $\frac{d}{d x}\left(\frac{u}{v}\right)$
3. True or False $\frac{d}{d x}\left(\pi^{3}\right)=3 \pi^{2}$. Justify your answer.

True or False The graph of $f(x)=1 / x$ has no horizontal tangents. Justify your answer.
4.

Let $f$ be the function defined as
$$
f(x)=\left\{\begin{array}{ll}
3-x, & x

Q: Homework8 Name: 1. Find the derivatives of the following functions using differentiation rules. a) $y=\frac{x^{3}}{3}-x$ b) $y=(x+1)\left(x^{2}+1\right)$ c) \[ y=\frac{x^{2}+5 x-1}{x^{2}} \] 2. Suppose $u$ and $v$ are functions of $x$ that are differentiable at $x=0$, and that $u(0)=5, u^{\prime}(0)=-3, v(0)=-1$, $v^{\prime}(0)=2$. Find the values of the following derivatives at $x=0$. (a) $\frac{d}{d x}(u v)$ (b) $\frac{d}{d x}\left(\frac{u}{v}\right)$ 3. True or False $\frac{d}{d x}\left(\pi^{3}\right)=3 \pi^{2}$. Justify your answer. True or False The graph of $f(x)=1 / x$ has no horizontal tangents. Justify your answer. 4. Let $f$ be the function defined as \[ f(x)=\left\{\begin{array}{ll} 3-x, & x<1 \\ a x^{2}+b x, & x \geq 1 \end{array}\right. \] where $a$ and $b$ are constants. (a) If the function is continuous for all $x$, what is the relationship between $a$ and $b$ ? (b) Find the unique values for $a$ and $b$ that will make $f$ both continuous and differentiable.

∻Solution by Steps∻ ‖ step 1 ⋮ To find the derivative of the function $ y = \frac{x^3}{3} - x $, we differentiate: $ \frac{d}{dx}\left(\frac{x^3}{3} - x\right) = x^2 - 1 $. ‖ ‖ step 2 ⋮ Setting the derivative equal to zero: $ x^2 - 1 = 0 $ gives us $ x = \pm 1 $ and $ x = 0 $. ‖ ‖ step 3 ⋮ Therefore, the critical points are $ x = 1, -1, 0 $. ‖ ‖ step 4 ⋮ For the function $ y = (x + 1)(x^2 + 1) $, we apply the product rule: $ \frac{d}{dx}((x + 1)(x^2 + 1)) = (x^2 + 1) + (x + 1)(2x) $. ‖ ‖ step 5 ⋮ Simplifying gives us $ 2x^2 + 2x + 1 $. Setting this equal to zero: $ 2x^2 + 2x + 1 = 0 $. ‖ ‖ step 6 ⋮ The solutions to this quadratic equation are $ x = -\frac{1}{3}i(\sqrt{2} + -i) $ and $ x = \frac{1}{3}i(\sqrt{2} + i) $, along with $ x = -\frac{2}{3} $ and $ x = \frac{1}{3} $. ‖ ∻Answer∻ ⚹ The critical points for the first function are $ x = 1, -1, 0 $ and for the second function, the solutions are $ x = -\frac{1}{3}i(\sqrt{2} + -i), x = \frac{1}{3}i(\sqrt{2} + i), -\frac{2}{3}, \frac{1}{3} $. ⚹ ∻Key Concept∻ ⚹Finding derivatives and critical points of functions.⚹ ∻Explanation∻ ⚹The derivatives help identify where the function's slope is zero, indicating potential maxima, minima, or points of inflection. The critical points are essential for analyzing the behavior of the function.⚹◊What differentiation rule would you apply to find the derivative of the function $y = \frac{x^{3}}{3} - x$?⍭ Generate me a similar question◊

Question

Homework8
Name:
1. Find the derivatives of the following functions using differentiation rules.
a) $y=\frac{x^{3}}{3}-x$
b) $y=(x+1)\left(x^{2}+1\right)$
c)
$$
y=\frac{x^{2}+5 x-1}{x^{2}}
$$
2. Suppose $u$ and $v$ are functions of $x$ that are differentiable at $x=0$, and that $u(0)=5, u^{\prime}(0)=-3, v(0)=-1$, $v^{\prime}(0)=2$. Find the values of the following derivatives at $x=0$.
(a) $\frac{d}{d x}(u v)$
(b) $\frac{d}{d x}\left(\frac{u}{v}\right)$
3. True or False $\frac{d}{d x}\left(\pi^{3}\right)=3 \pi^{2}$. Justify your answer.

True or False The graph of $f(x)=1 / x$ has no horizontal tangents. Justify your answer.
4.

Let $f$ be the function defined as
$$
f(x)=\left\{\begin{array}{ll}
3-x, & x<1 \
a x^{2}+b x, & x \geq 1
\end{array}\right.
$$
where $a$ and $b$ are constants.
(a) If the function is continuous for all $x$, what is the relationship between $a$ and $b$ ?
(b) Find the unique values for $a$ and $b$ that will make $f$ both continuous and differentiable.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the derivative of the function $ y = \frac{x^3}{3} - x $, we differentiate: $ \frac{d}{dx}\left(\frac{x^3}{3} - x\right) = x^2 - 1 $. ‖ ‖ step 2 ⋮ Setting the derivative equal to zero: $ x^2 - 1 = 0 $ gives us $ x = \pm 1 $ and $ x = 0 $. ‖ ‖ step 3 ⋮ Therefore, the critical points are $ x = 1, -1, 0 $. ‖ ‖ step 4 ⋮ For the function $ y = (x + 1)(x^2 + 1) $, we apply the product rule: $ \frac{d}{dx}((x + 1)(x^2 + 1)) = (x^2 + 1) + (x + 1)(2x) $. ‖ ‖ step 5 ⋮ Simplifying gives us $ 2x^2 + 2x + 1 $. Setting this equal to zero: $ 2x^2 + 2x + 1 = 0 $. ‖ ‖ step 6 ⋮ The solutions to this quadratic equation are $ x = -\frac{1}{3}i(\sqrt{2} + -i) $ and $ x = \frac{1}{3}i(\sqrt{2} + i) $, along with $ x = -\frac{2}{3} $ and $ x = \frac{1}{3} $. ‖ ∻Answer∻ ⚹ The critical points for the first function are $ x = 1, -1, 0 $ and for the second function, the solutions are $ x = -\frac{1}{3}i(\sqrt{2} + -i), x = \frac{1}{3}i(\sqrt{2} + i), -\frac{2}{3}, \frac{1}{3} $. ⚹ ∻Key Concept∻ ⚹Finding derivatives and critical points of functions.⚹ ∻Explanation∻ ⚹The derivatives help identify where the function's slope is zero, indicating potential maxima, minima, or points of inflection. The critical points are essential for analyzing the behavior of the function.⚹◊What differentiation rule would you apply to find the derivative of the function $y = \frac{x^{3}}{3} - x$?⍭ Generate me a similar question◊

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