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Tangent Lines and Their Slopes

Objectives

  • Relate the derivative, f ′ ( a ) , to the slope of the tangent line that contains the point ( a , f ( a ) ) .
  • Determine the equation of the tangent line that contains the point ( a , f ( a ) ) .

Summary

The tangent line to a differentiable function y = f ( x ) at the point ( a , f ( a ) ) is given in point-slope form by the equation

y − f ( a ) = f ′ ( a ) ( x − a ) .

The principle of local linearity tells us that if we zoom in on a point where a function y = f ( x ) is differentiable, the function will be indistinguishable from its tangent line. That is, a differentiable function looks linear when viewed up close. We rename the tangent line to be the function y = L ( x ) , where L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) . Thus, f ( x ) ≈ L ( x ) for all x near x = a .

If we know the tangent line approximation L ( x ) = f ( a ) + f ′ ( a ) ( x − a ) to a function y = f ( x ) , then because L ( a ) = f ( a ) and L ′ ( a ) = f ′ ( a ) , we also know the values of both the function and its derivative at the point where x = a . In other words, the linear approximation tells us the height and slope of the original function. If, in addition, we know the value of f ″ ( a ) , we then know whether the tangent line lies above or below the graph of y = f ( x ) , depending on the concavity of f .

See the Desmos Demonstration.

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