Equation of Tangent line

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NaazB_3.1415

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Hey I got this question in my calculus textbook, I've tried everything to solve it, can someone help me out? Here is the full question:
"There are 2 tangent lines to the curve y=4x - x^2 that pass through the point (2,5). Find the equations of those 2 lines and make a sketch to verify your results."
I know this problem may be rudimentary for some, but I can't seem to crack it. Thanks :)
 
Apparently, there is something you haven't tried.

Lines through (2,5) look like this: (y-5) = m(x-2)

Slope of tangent lines to y = 4x - x^2 look like this: m = 4 - 2x
 
NaazB_3.1415 said:
Hey I got this question in my calculus textbook, I've tried everything to solve it, can someone help me out? Here is the full question:
"There are 2 tangent lines to the curve y=4x - x^2 that pass through the point (2,5). Find the equations of those 2 lines and make a sketch to verify your results."
I know this problem may be rudimentary for some, but I can't seem to crack it. Thanks :)
Click to expand...
You said "I've tried everything to solve it,"

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
Subhotosh Khan said:
You said "I've tried everything to solve it,"

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
Click to expand...
I first reasoned that the slope of the tangent line must be a point common to the derivative of the original curve. Then I have to figure out at what X value that slope occurs at. Because the curve is a parabola I can assume that the other tangent occurs at the same y coordinate but has the negative slope of the first tangent. As you can see in my work, the equation has more than one unknown so I get stuck there, leading me to doubt my approach to the question overall.
 
NaazB_3.1415 said:
Here is the full question: "There are 2 tangent lines to the curve y=4x - x^2 that pass through the point (2,5). Find the equations of those 2 lines and make a sketch to verify your results."
Click to expand...
Every point on the curve looks like \((x,4x-x^2)\) and the slope of the tangent at any of those points is \(4-2x\).
If a tangent line passes through \((2,5)\) its slope must be \(\dfrac{5-(4x-x^2)}{2-x}=~?\) This may help you.
 
Slope-Intercept is just one way to express the equation of a line. Please learn, and be fluent with, the many other forms. Just one will not serve you well in the long run.
 
OH thanks, that's really good advice, I often find myself and observe others as well, who can answer a question when its formatted a certain way, but have difficulty answering a question based on the same fundamental concepts but written in a different form, or needs solving from an angled approach.
 
pka said:
Every point on the curve looks like \((x,4x-x^2)\) and the slope of the tangent at any of those points is \(4-2x\).
If a tangent line passes through \((2,5)\) its slope must be \(\dfrac{5-(4x-x^2)}{2-x}=~?\) This may help you.
Click to expand...
Thanks this helped me solve it, I also learned from this the power of the slope formula in conjunction with the derivative of a function!
 
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