Question

A point P(x,y) moves along the graph of the equation y = x3 + x2 + 2. The x-values are changing at the rate of 2 units per second. How fast are the y-values changing (in units per second) at the point Q(1,4)?

Answer 1

Using implicit differentiation, it is found that y is changing at a rate of 10 units per second.

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The equation is:

$y = x^3 + x^2 + 2$

Applying implicit differentiation in function of t, we have that:

$\frac{dy}{dt} = 3x^2\frac{dx}{dt} + 2x\frac{dx}{dt}$

• x-values changing at a rate of 2 units per second means that $\frac{dx}{dt} = 2$
• Point Q(1,4) means that $x = 1, y = 4$.

We want to find $\frac{dy}{dt}$, thus:

$\frac{dy}{dt} = 3x^2\frac{dx}{dt} + 2x\frac{dx}{dt}$

$\frac{dy}{dt} = 3(1)^2(2) + 2(1)(2) = 6 + 4 = 10$

y is changing at a rate of 10 units per second.

A similar problem is given at brainly.com/question/9543179