Exploring Tangent Lines in Calculus Problems

In summary, the conversation discusses the connection between tangent lines and the given problem. The derivative of a function is the slope of the tangent line at a specific point. To find the value of the function at x=7, F(7), you isolate y in the equation and substitute 7 for x. To find the derivative at x=7, F'(7), you first differentiate the equation and then plug in 7 for x. Differentiating means finding the derivative of a function.
  • #1
ACLerok
194
0
how does tangent lines play into this problem?

http://www.eden.rutgers.edu/~cjjacob/images/calc.jpg [Broken]
 
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  • #2
Do you understand that f(7) is the value of y when x=7?
 
  • #3
yeah. i don't understand F'(7).
 
  • #4
dy/dx = f'(x) is the slope of the curve y = f(x), so what is f'(7)?
 
  • #5
am i supposed to plug 7 in for x and then take the derivative? how? doesn't the equation end up canceling out?
 
  • #6
Why would you take the derivative?

The slope of the tangent line = the slope of the curve at the point where the line touches the curve.
 
  • #7
oh, i didnt realize the d's stood for the difference.
so to find F(7), I isolate y on one side of the equation and then plugin 7 for x and solve for y? what do i do for the F'(7)? slope of the line when x=7?
 
  • #8
Originally posted by ACLerok
oh, i didnt realize the d's stood for the difference.
so to find F(7), I isolate y on one side of the equation and then plugin 7 for x and solve for y? what do i do for the F'(7)? slope of the line when x=7?

These are all things that should be elementary (believe me, the problems are going to get a lot harder!).

In the first place, the "d's" do not stand for difference!
They are simply the notation for derivative. Yes, one method of finding the derivative of a function is to take the limit of the "difference quotient" but I don't think you should think "the d's stand for the difference".

In any case, the first thing you should have learned about the derivative of a function is that "the derivative IS the slope of the tangent line".

In this case, yes, solve for y. The value of y when x= 7 is the value of the function F when x= 7, F(7). The slope of that line is the derivative of F when x= 7 dF/dx(7) or F'(7).

By the way, you do not find the derivative of a function "when x= 7" by substituting 7 for x and then differntiating. The value of any function for a specific x is a number (a constant) and the derivative of a constant is always 0. Do it the other way around: first differentiate and then substitute.
 
  • #9
so correct me if wrong. to find F(7), i just plug 7 for x in the equation of the tangent line and to find F'(7), I just differentiate the equation and then plug in 7 for x and solve for y?

I forget, what does differentiate mean?
 

What is a tangent line in a calculus problem?

A tangent line is a line that touches a curve at only one point, called the point of tangency. It represents the instantaneous rate of change of the curve at that point.

How do you find the equation of a tangent line?

To find the equation of a tangent line, you need to first find the slope of the tangent line by taking the derivative of the original function at the point of tangency. Then, you can use the point-slope formula to find the equation of the tangent line.

Why are tangent lines important in calculus?

Tangent lines are important in calculus because they allow us to find the instantaneous rate of change of a function at a specific point. This is useful in many real-world applications, such as finding the velocity of an object at a given time or the rate of change of a chemical reaction at a specific moment.

How do you determine if a line is tangent to a curve?

A line is tangent to a curve if it touches the curve at only one point and has the same slope as the curve at that point. This can be determined by graphing the line and the curve and visually checking for the point of tangency.

What are some common applications of tangent lines in real life?

Tangent lines have many real-life applications, such as in physics, where they can be used to find the velocity and acceleration of an object at a specific time. In economics, they can be used to analyze the marginal cost and revenue of a business. In engineering, they are used to design smooth curves for roads and bridges. Tangent lines also have applications in medicine, astronomy, and many other fields.

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