High-speed elevators function under two limitations: (1) the maximum magnitude of vertical acceleration that a typical human body can experience without discomfort is about $1.2 \mathrm{~m} / \mathrm{s}^2$, and (2) the typical maximum speed attainable is about $9.0 \mathrm{~m} / \mathrm{s}$. You board an elevator on a skyscraper's ground floor and are transported $180 \mathrm{~m}$ above the ground level in three steps: acceleration of magnitude $1.2 \mathrm{~m} / \mathrm{s}^2$ from rest to $9.0 \mathrm{~m} / \mathrm{s}$, followed by constant upward velocity of $9.0 \mathrm{~m} / \mathrm{s}$, then deceleration of magnitude $1.2 \mathrm{~m} / \mathrm{s}^2$ from $9.0 \mathrm{~m} / \mathrm{s}$ to rest. What fraction of the total transport time does the normal force not equal the person's weight?

A student forgot the Product Rule for differentiation and made the mistake of thinking that $(fg)' = f'g'$. However, he was lucky and got the correct answer. The function $f$ that he used was $f(x) = e^{x^2}$ and the domain of his problem was the interval $(\frac{1}{2}, \infin)$. What was the function $g$?

Find dy/dx by implicit differentiation. $e^{x-y}=\ln (x-y)$

Find dy/dx by implicit differentiation. $y^{2}+y=\ln x$

Use complete sentences to answer the following questions. Where does the process of digestion begin, and which digestive juice is involved?

Find $dy/dx$ by implicit differentiation.

$$y^2=(x-y)(x^2+y)$$

Differentiate $\left(x^2-5 x+8\right)\left(x^3+7 x+9\right)$ in three ways mentioned below:

(i) by using product rule

(ii) by expanding the product to obtain a single polynomial.

(iii) by logarithmic differentiation.

Do they all give the same answer?

Use implicit differentiation to find dy/dx. x = sec y

$$\begin{array} { l } { \text {Extend the Product Rule for differentiation to the follow- } } \\ { \text {ing case involving the product of three differentiable } } \\ { \text {functions: Let } h ( x ) = u ( x ) v ( x ) w ( x ) , \text { and show that } } \\ { h ^ { \prime } ( x ) = u ( x ) v ( x ) w ^ { \prime } ( x ) + u ( x ) v ^ { \prime } ( x ) w ( x ) + u ^ { \prime } ( x ) v ( x ) w ( x ) } \\ { \text {Hint: Let } f ( x ) = u ( x ) v ( x ) , g ( x ) = w ( x ) , \text { and } h ( x ) = f ( x ) g ( x ) , \text {} } \\ { \text {and apply the Product Rule to the function $h$. } } \end{array}$$

Find $dy/dx$ by implicit differentiation. $e^{x/y} = 5x − y$.

Use logarithmic differentiation to compute f'(x)/f(x) where $f(x)=(x)^{x^2}$.

Imaginez que vous etes eleve dans un lycee parisien.

Un samedi, vous voulez aller a la Cite des Sciences.

• Quel transport public est-ce que vous prenez?

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• A quelle station est-ce que vous descendez?

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Some hotels in Las Vegas use moving sidewalks to transport guests between various destinations. Suppose a hotel guest decides to get on a moving sidewalk that is $195 \mathrm{~ft}$ long. If the guest walks at a rate of $5 \mathrm{~ft} / \mathrm{s}$ and the moving sidewalk moves at a rate of $7 \mathrm{~ft} / \mathrm{s}$, how many seconds will it take the guest to walk from one end of the moving sidewalk to the other?

Find $dy/dx$ using implicit differentiation

$$xe^y= x-y$$