Try It Yourself#
Exercise 1#
Implicit differentiation
Find \(\dfrac{dy}{dx}\) where \(y\) is defined implicitly by \(y^4 + x^3y^2 - x = 5\).
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Answer: \((1 - 3x^2y^2)/(4y^3 + 2x^3y)\)
Exercise 2#
Equation of tangent line via implicit differentiation
Find the equation of the line tangent to the curve defined implicitly by \(x^4y-x^3+14y=29\) at the point \((1,2)\).
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Answer: \(y=-x/3 + 7/3\)
Exercise 3#
Elasticity of demand using implicit differentiation
The demand equation for Hager & Little’s Better Business Calculus book series is given by
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 \(x=4\).
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Answer: \(1/14\)
Exercise 4#
Related rates problem: demand vs. revenue
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
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.
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Answer: revenue increases by $3,500 per day
Exercise 5#
Related rates problem: price vs. demand
If the quantity demanded daily of a product is related to its unit price in dollars by
how fast is the quantity demanded changing when \(x=5\) and the unit price is decreasing at a rate of $3 per day?
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Answer: demand increases by \(27/5\) units per day
Exercise 6#
Related rates problem: price vs. demand
The weekly demand function is given by
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?
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Answer: demand decreases by 260 units per week