Outcome A4: Implicit Curves and Linearization

Question 1

What is an explicit function?

Answer 1

A function in which the dependent variable can be written explicitly in terms of the independent variable.

Question 2

What is an implicit function?

Answer 2

An implicit function is a function whose inputs and outputs satisfy a given equation. It is not possible to isolate one variable to one side.

Question 3

How can an implicit function be graphed?

Answer 3

If we were to plot all the possible points (x, y) satisfying the equation, we would find that some x-values correspond with multiple y-values. Thus, we must split the graph into three pieces, each of which define the function.

Question 4

How can we find the slope at any point for an implicit function?

Answer 4

We cannot find an explicit formula for any portion of the function. But, if (x, f(x)) is an input-output pair of any of these functions, we know that (x, f(x)) satisfies the implicit function equation such that we can replace y with f(x). This will allow us to find the slope of f(x) at any given point (x, f(x)).

Question 5

How do we approach implicit differentiation?

Answer 5

Given an equation in x and y defining an implicit function, we can find dy/dx by taking the derivative with respect to x, keeping in mind that y is an implicit function of x.

Question 6

What is the formula for point-slope form?

Answer 6

y = m(x-x0) + y0

Question 7

If we are looking for where a function has a horizontal tangent, what should we set the slope equal to?

Answer 7

Set the derivative equal to 0. Once you find the x-values that fit, make sure that slope exists by seeing if the x-value results in a value of 0 in the denominator.

Question 8

What is linearization (linear approximation)?

Answer 8

If a function is differentiable at a point x = a, it looks like a line near x = a. So, if we define the linear equation L(x) = f’(a)(x-a) + f(a) (which is the equation of the tangent line), then f(x) ~ L(x) for x near a.

Question 9

How should we find a function f(x) and an input b?

Answer 9

Find a function f(x) and an input b such that f(b) is the quantity you’re trying to approximate

Question 10

How should we determine a?

Answer 10

Find an input a as close to b as possible such that f(a) and f’(a) are easy to calculate

Question 11

What are the steps to approaching a linearization problem?

Answer 11

1. Set the value you are trying to approximate equal to the original function to find the x-value and then find a value a that is close to that x-value.
2. Take the derivative of the original function.
3. Plug in a to the original function, derivative, and line equation to get the point-slope form.
4. Use the x-value found earlier and plug into L(x)