If the gradient of the tangent to a curve is given by $2x+1$ and the curve passes through the point $(-3,0)$, find the equation of the curve.
Shouldn't it be the equation of the tangent to the curve since the gradient in a linear equation of the form $ax+b$ is represented by $a$?
Best Answer
You're given: $\frac{dy}{dx}=2x+1$
By simple integration you obtain:
$$y=x^2+x-6$$
Now, you say that:
This is obviously a misunderstood claim. The expression $2x+1$ gives you the slope of the tangent at a point $x$ of the graph, it does NOT give you the equation of the tangent at that point.
For a comparison, at point $P(-3,0)$, the slope of the tangent is $m=2\cdot(-3)+1=-5$ and so, the equation of the tangent is $y-0=-5\cdot(x-(-3))$. You can verify it from graph as well:
Also, note that different points of the curve will have different slopes and correspondingly different equations for tangents.
The line $y=2x+1$, thus, is in no way a "tangent" to the graph of $y$.
Hope your query is clarified!