AP Calculus AB · Differentiation definitions and fundamental rules
The curve x^2+3xy+2y^2=12 contains the point (2,1)
Problem
The curve \(x^2+3xy+2y^2=12\) contains the point \((2,1)\). Treating \(y\) locally as a differentiable function of \(x\), (a) differentiate the equation implicitly and solve for \(\dfrac{dy}{dx}\); (b) evaluate \(\dfrac{dy}{dx}\) at \((2,1)\); (c) find an equation of the tangent line to the curve at \((2,1)\). Show the product-rule term that arises from \(xy\).
Hint
Treat \(y\) as a function of \(x\) and differentiate every term, using the product rule on \(xy\).
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free AP Calculus AB sample problem: Try the free sample problem.
More AP Calculus AB practice problems
- On the interval [-4,6], the graph of a function r consists of three line segments: the…Limits and continuity
- A continuous function T gives the outdoor temperature in degrees Fahrenheit t hours…Limits and continuity
- A drone flies horizontally at a constant altitude of 80 metersDifferentiation definitions and fundamental rules
- For 0<x<π, let y=(sin x)^xComposite, implicit, and inverse-function differentiation
- The cost, in dollars, of producing q liters of a chemical is given by a differentiable…Contextual applications of differentiation
- A closed right circular cylinder must enclose a volume of 250π cubic centimetersAnalytical applications of differentiation
- A well-stirred tank initially contains 80 liters of pure waterIntegration and accumulation of change
- A tank contains 100 gallons of water at time t=0 minutesDifferential equations
- Let R be the region bounded by y=4-x^2 and y=x+2Applications of integration