Derivatives · Definition
Tangent line
Before you start you need to know
A tangent line approximates the graph of \(f\) near a point, and its slope is the derivative at that point.
Definition
The tangent line to the graph of \(f\) at the point \((a,f(a))\) is the line through that point with slope \(f^{\prime}(a)\). It is the best linear approximation of \(f\) near \(a\).
The tangent line at \((a,f(a))\) is the line that best approximates the graph near that point. Its slope is \(f'(a)\), so its equation is \(y=f(a)+f'(a)(x-a)\).
For example, for \(f(x)=x^{2}\) at \(a=1\): \(f(1)=1\), \(f'(1)=2\), and the tangent is \(y=2x-1\).
Typical error: using \(f'(x)\) instead of \(f'(a)\), or writing \(y=f'(a)(x-a)\) without adding \(f(a)\).
Think about it
Find the tangent to \(y=x^2\) at \(x=1\).
f(1)=1 and f’(1)=2, so y = 1 + 2(x−1), that is, y = 2x − 1.
Exercises 10
- Find the equation of the tangent line to \(f(x)=x^{2}\) at \(x=1\).
- Find the equation of the tangent line to \(f(x)=x^{3} - 2 x\) at \(x=2\).
- Find the equation of the tangent line to \(f(x)=\sqrt{x}\) at \(x=4\).
- Find the equation of the tangent line to \(f(x)=\frac{1}{x}\) at \(x=1\).
- Find the equation of the tangent line to \(f(x)=e^{x}\) at \(x=0\).
- Find the equation of the tangent line to \(f(x)=\ln{\left(x \right)}\) at \(x=1\).
- What is the slope of the tangent line to \(f\) at \(x=a\)?
- Through which point must the tangent line to \(f\) at \(x=a\) pass?
- Find the equation of the tangent line to \(f(x)=x^{2}\ln(x)\) at \(x=1\).
- What is the tangent line to the line \(y=5x-2\) at any point on it?
References
- James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.