I have a math problem that I am a little "iffy" on.
1: Find the slope of the curve at point $P(1,4)$ and find the equation of the tangent line at P.
$y=5-x^2 P(1,4)$.
Now to find the slope, is $(y_2-y_1)/(x_2-x_1)$, but to do this, I need to know point Q. But I am not 100% sure I found Q correctly. I got $Q(1+h),(2+h)^2$ but I am a little stuck there.
So
$(2+h)^2-1^2/h$ would be the equation to find the slope? Once I know the slope I think I can just plug and chug point slope formula for the tangent line using P and the slope.
Thank you
$\endgroup$ 22 Answers
$\begingroup$To find the slope of the curve, or the gradient you can differentiate the function and plug in the $x$ value of the point into the derivative and that will yield the gradient at that point. So you have $y=5-x^2$ at the point $(1, 4)$. $$y'=-2x$$ The gradient of the curve, $y=5-x^2$ at $(1,4)$ is: $$y'(1)=-2(1)=-2$$
For the equation of the tangent line, you have the gradient $(-2)$, a point on the line $(1, 4)$, and you know that the equation of a line is $y=mx+c$. You can plug in the values that you know to find the value of $c$ and then you'll have the equation of the tangent line.
$\endgroup$ 6 $\begingroup$Since $$y=5-x^2\\ \frac{dy}{dx}=-2x$$ This gives for the tangent line: $$y-y_1=\frac{dy}{dx}(x-x_1)\\ =y-y_1=2x(x_1-x)$$ At $P=(1,4)$, this is $$y-4=2x(1-x)\\ \boxed{y-4=2x-2x^2}$$
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