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When learning about the concept of the derivative, we often refer to the derivative as the slope of the tangent line.
Discussion
The line that is tangent to f(x) at x=a touches the curve of f at x=a and has the same slope.
Write an equation for the tangent line at x=a in terms of f and f′.
The tangent to y=f(x) at x=a has slope f′(a) and passes through (a,f(a)). Hence in point–slope form
or equivalently
Tangent to f(x)
L:mx+c is tangent to f(x) at x=a means
Using point slope form the equation of the tangent is:
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Exercise
A function f has f(0)=5 and f′(0)=3. Find the equation y=mx+c of the tangent to the curve y=f(x) at x=0.
Select the correct option
Exercise
Let f(x)=x3−x2.
Find the equation of the tangent line to the curve y=f(x) at x=1 in the form y=mx+c.
Select the correct option
Discussion
The normal line at x=a is perpendicular to the tangent and intersects with the tangent at x=a. Find an expression for the normal line in terms of f and f′.
The tangent to y=f(x) at x=a has slope f′(a), so the normal (being perpendicular) has slope −1/f′(a). Since it passes through (a,f(a)), its equation is
Hence,
Normal to f(x)
The normal to f(x) at x=a is the line that passes through (a,f(a)) and is perpendicular to the tangent:
Using point slope form the equation of the tangent is:
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Exercise
Given that a function f has f(−2)=−2 and f′(−2)=3, find the equation of the normal to the curve y=f(x) at x=−2 in the form y=mx+c.
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Exercise
Let f(x)=x3−2x.
Find the equation of the normal to the curve y=f(x) at the point where x=−1, giving your answer in the form y=mx+c.
Select the correct option
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