Exercise · ZW-TANGENT-LINE-004 · ★★☆ Intermediate
Find the equation of the tangent line to \(f(x)=\frac{1}{x}\) at \(x=1\).
Choose one option.
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- D. \(y = 2 - x\)
Explanation
Compute \(f(1)=1\) and \(f'(x)=- \frac{1}{x^{2}}\), so \(f'(1)=-1\). The line is \(y-f(1)=f'(1)(x-1)\), that is, \(y = 2 - x\).
Review the concept: Tangent line
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 \(a\).
\[y-f(a)=f'(a)\,(x-a)\]
More practice problems: tangent line
- ZW-TANGENT-LINE-001 Find the equation of the tangent line to f(x)=x^2 at x=1.
- ZW-TANGENT-LINE-002 Find the equation of the tangent line to f(x)=x^3 - 2 x at x=2.
- ZW-TANGENT-LINE-003 Find the equation of the tangent line to f(x)=√x at x=4.
- ZW-TANGENT-LINE-005 Find the equation of the tangent line to f(x)=e^x at x=0.
- ZW-TANGENT-LINE-006 Find the equation of the tangent line to f(x)=ln (x ) at x=1.
- ZW-TANGENT-LINE-007 What is the slope of the tangent line to f at x=a?
- ZW-TANGENT-LINE-008 Through which point must the tangent line to f at x=a pass?
- ZW-TANGENT-LINE-009 Find the equation of the tangent line to f(x)=x^2ln (x) at x=1.