(a) find two explicit functions by solving the equation for...

(a) find two explicit functions by solving the equation for in terms of (b) sketch the graph of the equation and label the parts given by the corresponding explicit functions, (c) differentiate the explicit functions, and (d) find $d y / d x$ implicitly and show that the result is equivalent to that of part (c). $x^{2}+y^{2}=64$

Key Concepts

Solving for Explicit Functions

This concept involves rewriting an equation given in implicit form into one or more explicit forms by isolating one variable in terms of the other. This is particularly useful when the implicit relationship defines more than one branch or portion of a curve, allowing each branch to be treated separately as a function with a well-defined input-output rule.

Graphing Implicit Curves

Graphing implicit curves involves plotting the set of points that satisfy an equation without first solving for one variable in terms of the other. Understanding how to break the graph into different segments or branches, especially when explicit representations exist, is key to visualizing the overall structure of the curve, such as symmetry and boundedness.

Differentiating Explicit Functions

This key concept refers to the process of finding the derivative of a function when it is expressed explicitly, typically using standard rules of differentiation. Calculating the derivative of explicit functions allows for the determination of slopes, rates of change, and can provide insights into the behavior of different segments of the curve.

Implicit Differentiation

Implicit differentiation is used to differentiate equations where the dependent and independent variables are intertwined and not easily separable. By differentiating both sides of the equation with respect to the independent variable and then solving for the derivative of the dependent variable, one can obtain the slope of the curve at any point, which should agree with the derivative obtained from the explicit forms when available.

Transcript

00:01 Problem 17.

00:02 X squared plus y squared equals 64.

00:05 So the first thing we should notice is that this is our equation of an ellipse.

00:11 So we should keep that in mind.

00:14 So it's a four -part question.

00:16 They ask us to put this in y equals form.

00:18 So let's do that.

00:22 Y squared equals 64 minus x squared.

00:26 And then we can take the square root of our function, y equals plus or minus.

00:36 64 minus x squared.

00:40 That is our y equals an x terms equation.

00:49 Next they ask us to graph it.

00:53 And what we should notice about this, like i said, this is our equation of an ellipse.

00:59 And since it's plus or minus right here, this is going to end up being an ellipse on the top.

01:09 And this should be a label it first.

01:17 Negative 8, negative 8, 0.

01:28 And it's going to be touching at those points, and then since it's negative as well, we'll be touching at these points.

01:35 And that should be more circular, but i think you get the idea.

01:40 And if you want to verify this, you can make an xy table and just verify all of these.

01:48 So like if we plug in 0, we would get plus or minus 8, and so on...

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