How Can I Simplify This Tangent Line Equation?

In summary, we are given a function and asked to compute the derivative at a given point. Using the definition of the derivative and the point-slope formula, we can find the tangent line at the given point. An interactive graph is available to visualize the tangent lines for different values of $b$. In this specific problem, we are given $b=4$ and the tangent line is given by $y=-\frac{3}{16}x+\frac{3}{2}$.
  • #1
Teh
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  • #2
We are given the function:

\(\displaystyle f(x)=\frac{3}{x}\)

Using the given definition, we are to compute:

\(\displaystyle f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}\frac{\dfrac{3}{x}-\dfrac{3}{a}}{x-a}=\lim_{x\to a}\frac{3a-3x}{ax(x-a)}=-\lim_{x\to a}\frac{3(x-a)}{ax(x-a)}=-\lim_{x\to a}\frac{3}{ax}=-\frac{3}{a^2}\)

And so the tangent line at the point $(b,f(b))$ would be given by (using the point-slope formula):

\(\displaystyle y=-\frac{3}{b^2}(x-b)+\frac{3}{b}=-\frac{3}{b^2}x+\frac{6}{b}\)

Here's an interactive graph to show you the functions and some of its tangent lines...you can use the "slider" for $b$ to see the tangent line for different values of $b$.

[desmos=-5,-5,5,5]y=\frac{3}{x};y=-\frac{3}{b^2}x+\frac{6}{b};b=1[/desmos]

In this problem, we are given $b=4$...so what is the tangent line?
 
  • #3
MarkFL said:
We are given the function:

\(\displaystyle f(x)=\frac{3}{x}\)

Using the given definition, we are to compute:

\(\displaystyle f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a}\frac{\dfrac{3}{x}-\dfrac{3}{a}}{x-a}=\lim_{x\to a}\frac{3a-3x}{ax(x-a)}=-\lim_{x\to a}\frac{3(x-a)}{ax(x-a)}=-\lim_{x\to a}\frac{3}{ax}=-\frac{3}{a^2}\)

And so the tangent line at the point $(b,f(b))$ would be given by (using the point-slope formula):

\(\displaystyle y=-\frac{3}{b^2}(x-b)+\frac{3}{b}=-\frac{3}{b^2}x+\frac{6}{b}\)

Here's an interactive graph to show you the functions and some of its tangent lines...you can use the "slider" for $b$ to see the tangent line for different values of $b$.
In this problem, we are given $b=4$...so what is the tangent line?

the tangent line will be 3/2
 
  • #4
Teh said:
the tangent line will be 3/2

The will be the $y$-intercept, but not the line itself. :D
 
  • #5
$$\displaystyle f(x)= -\frac{3}{16} x+\frac{3}{2}$$
 

FAQ: How Can I Simplify This Tangent Line Equation?

What are the limits of tangent lines?

The limits of tangent lines refer to the values that a tangent line approaches as it gets closer and closer to a specific point on a curve. It represents the slope of the curve at that point and can be used to find the instantaneous rate of change.

How are limits of tangent lines calculated?

Limits of tangent lines can be calculated using the derivative of the function at the given point. This derivative represents the slope of the tangent line at that point.

Why are limits of tangent lines important in calculus?

Limits of tangent lines are important in calculus because they allow us to determine the behavior of a function as it approaches a specific point. This is crucial in finding the instantaneous rate of change, which is used in many real-world applications.

What does a limit of a tangent line tell us about a curve?

The limit of a tangent line tells us the slope of the curve at a specific point. It also gives us information about the direction and steepness of the curve at that point.

Are there any practical applications of limits of tangent lines?

Yes, there are many practical applications of limits of tangent lines. For example, they are used in physics to calculate the velocity and acceleration of an object at a specific point in time. They are also used in economics to determine the marginal cost and revenue of a product at a given quantity.

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