Try It Yourself

Try It Yourself#

Exercise 1#

Find $\dfrac{dy}{dx}$ where $y$ is defined implicitly by

\[y^4 + x^3y^2 - x = 5.\]

Exercise 2#

Find the equation of the line tangent to the curve defined implicitly by

\[x^4y-x^3+14y=29\]

at the point $(1,2)$.

Exercise 3#

The demand equation for Hager & Little’s Better Business Calculus book series is given by

\[p = -5x^2 - 2x + 100\]

where $p$ is the wholesale unit price in dollars and $x$ is the quantity demanded in units of a thousand. Compute the elasticity of demand when $x=4$.

Exercise 4#

A company is increasing production of math-brain protein bars at a rate of 50 cases per day. All cases produced can be sold. The daily demand function is given by

\[p(x) = 80 - \frac{x}{100}\]

Where $x$ is the number of cases produced and sold, and $p$ is in dollars. Find the rate of change of the revenue with respect to the time in days when daily production is 500 cases.

Exercise 5#

If the quantity demanded daily of a product is related to its unit price in dollars by

\[p^2 = 106 - x^2\]

How fast is the quantity demanded changing when $x=5$ and the unit price is decreasing at a rate of $3 per day.

Exercise 6#

The weekly demand function is given by

\[p + x + 3xp = 43\]

where $x$ is the number of thousands of units demanded weekly and $p$ is in dollars. If the price is increasing at a rate of 20 cents per week when the level of demand is 4000 units, at what rate is the demand decreasing?