To clarify, the above should beJustabeginner said:Homework Statement
Find the equation of the tangent line at the curve y= x * e^(2x) at the point (2, 2*e^(4))
Homework Equations
The Attempt at a Solution
f'(x)= e^(2x) * (2x+ 1)
(e^4)(5) = 5e^4
Justabeginner said:y- 2*e^4 = 5*e^4 (x-2)
y= 5(e^4)*x - 8*(e^4)
Is this right? It seems too easy for me to have gotten it right :/ Thanks.
A tangent line at a curve is a line that touches the curve at exactly one point, without intersecting or crossing over it. It represents the slope of the curve at that specific point.
In calculus, the tangent line is used to find the slope or rate of change of a curve at a specific point. It is also used to find the equation of a curve and to approximate the value of a function at a given point.
The tangent line is calculated by finding the derivative of the curve at the given point. The derivative represents the slope of the curve at that point, so the equation of the tangent line can be written in the form y=mx+b, where m is the slope and b is the y-intercept.
A secant line is a line that intersects a curve at two or more points, while a tangent line only touches the curve at one point. The secant line can be thought of as the average slope between two points on the curve, while the tangent line represents the instantaneous slope at a single point.
No, a curve can only have one tangent line at a point. This is because the tangent line represents the slope of the curve at that point, and a curve can only have one slope at a specific point.