find an equation for the line tangent to the graph of y=arccot(x) at x=1.
I found that slope is -1/(1+x^2). I don't know how to evaluate arccot(1) to find y value.
arccot(1) means give me the value or the angle so that cot(that value) = 1
we know that tan(π/4) = 1 , (tan 45° = 1)
and since cotØ = 1/tanØ
arccot(1) = π/4 , thus y = π/4
so now you have the slope and the point (1,π/4)
carry on .....
Thank you very much! I get it now. For another question:
y'=lim h->0 ((e^x+h)-e^x)/h. So I simplified this and got (e^x)* lim h->0 ((e^h)-1)/h. But I don't understand why and how ((e^h)-1)/h equals 1. Explain?
if y = e^x
the derivative of y in terms of fundamental concepts or sometimes called by "first principles"
lim ( e^(x+h) - e^x)/h as h --->0 , (notice the proper placement of brackets)
so you are simply given the dy/dx of e^x
thus y = e^x
To find the equation for the line tangent to the graph of y = arccot(x) at x = 1, we need to determine both the slope of the tangent line and the y-coordinate of the point of tangency.
First, let's start by calculating the slope of the tangent line. The slope of the tangent line can be found by taking the derivative of the function y = arccot(x).
Differentiating y = arccot(x) with respect to x, we get:
dy/dx = -1/(1 + x^2)
Now, let's find the value of x at which the tangent line is to be drawn, which is x = 1 in this case.
Next, let's determine the y-coordinate of the point of tangency. To do this, we need to evaluate the function y = arccot(x) at x = 1.
To evaluate arccot(1), we know that arccot(x) is the angle whose cotangent equals x. So, arccot(1) is the angle whose cotangent is equal to 1. In this case, arccot(1) = π/4 or 45 degrees.
Now, we have the x-coordinate (x = 1) and the y-coordinate (y = π/4) of the point of tangency.
Since we know both the slope (-1/(1 + x^2)) and a point (1, π/4) on the tangent line, we can proceed to find the equation of the tangent line using the point-slope form.
Let's use the point-slope form: y - y1 = m(x - x1), where (x1, y1) represents the point of tangency and m represents the slope.
Substituting the values, we get:
y - π/4 = -1/(1 + 1^2)(x - 1)
Simplifying,
y - π/4 = -1/2(x - 1)
This can be further simplified to obtain the equation for the tangent line.