How do you find the original function given the equation of a tangent line and two points?
Well, finding the original function is like finding a needle in a haystack, except the haystack is filled with other needles and someone forgot to label them. It's quite the challenge! But fear not, I'm here to help, or at least entertain you while you search.
To find the original function given the equation of a tangent line and two points, we need a bit of detective work. Start by figuring out the slope of the tangent line using the two points. You can call that slope "Sherlock."
Once you have the slope, it's time to put on your mathematician hat and integrate it. Integration is like solving a puzzle, but instead of putting the pieces together, you're finding that missing function.
Now, here comes the fun part! Add a constant of integration, which you can think of as the wild card in this mathematical game. This constant saves the day when you think you've got all the answers but realize you missed something. It's the unexpected twist in a novel that keeps you guessing till the end.
And voila! You've got the original function, like finding the long-lost treasure hidden in a complicated math problem. So, grab that magnifying glass, put on your detective hat, and get ready for some mathematical mischief! Happy solving, my friend!
To find the original function given the equation of a tangent line and two points, you can follow these steps:
Step 1: Start with the equation of the tangent line. The equation of a tangent line can be written in the slope-intercept form as y = mx + b, where m is the slope and b is the y-intercept.
Step 2: Determine the slope of the tangent line. The slope of the tangent line is equivalent to the derivative of the original function at the point of tangency. You might have been given the slope directly or can calculate it by substituting the given x-value into the derivative equation.
Step 3: Use the point-slope form to write the equation of the tangent line. With the slope (m) and one point on the line (either of the two given points), you can substitute these values into the point-slope form, y - y1 = m(x - x1), to get the equation of the tangent line.
Step 4: Integrate the equation of the tangent line to find the original function. Integration is the reverse process of differentiation. By integrating the equation of the tangent line with respect to x, you can find the original function. This can involve adding a constant of integration (C) since indefinite integration yields a family of functions.
Step 5: Use the given point(s) to determine the value of the constant of integration. Substitute the x and y values of either of the given points into the equation obtained in step 4. This will allow you to find the specific value of the constant of integration and derive the particular original function.
That's it! Following these steps, you can find the original function given the equation of a tangent line and two points.
To find the original function given the equation of a tangent line and two points, you need to follow a systematic process. Here's a step-by-step explanation:
Step 1: Obtain the equation of the tangent line. The equation of the tangent line is typically given in the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) represents a point on the line and m represents the slope of the line.
Step 2: Determine the slope of the tangent line. The slope of the tangent line can be directly obtained from the given equation. If the equation is not given in the point-slope form, you may need to rearrange it to extract the slope.
Step 3: Find the derivative of the original function. The derivative gives the slope of the tangent line at any given point on the curve. By finding the derivative, you can find the slope of the tangent line at the given point.
Step 4: Use the two points given to setup a system of equations. Since you have two points, you can use them to create two equations. The first equation will involve the coordinates of one of the points and the original function. The second equation will involve the coordinates of the other point and the original function.
Step 5: Solve the system of equations. Solve the system of equations to find the values of the constant terms in the original function. This will give you the complete equation of the original function.
Step 6: Verify the obtained function. Substitute the x-values of the given points into the obtained function and check if the corresponding y-values match the given points. If they do, you have found the correct original function.
By following these steps, you should be able to find the original function given the equation of a tangent line and two points.
No way to tell, unless you know you are working with something simple, like a quadratic. If
y = ax^2+bx+c
and you have two points, then you also have
2ax+b = slope of the tangent line
Then you have three equations to solve for a,b,c.