What is the equation of the tangent line of #f(x)=(x-1)^3+(x-2)(x-7) # at #x=7#?

Answer 1

#y=113x-545#

Step 1: Determine the y coordinate at the point #x=7#
#f(x)=(x-1)^3+(x-2)(x-7)# #f(7)=(7-1)^3+(7-2)(7-7)rArr6^3+(5*0)rArr216# Therefore the tangent to the curve passes through the point #color(orange)((7;216))#

Step 2: Make the provided Function larger and simpler

#f(x)=(x-1)^3+(x-2)(x-7)# #rArrx^3-3x^2+3x-1+x^2-7x-2x+14# #rArrx^3-2x^2-6x+13#

Locate the Derivative in Step 3

#f'(x)=3x^2-2*(2x)-6# #rArr3x^2-4x-6#
Step 4: Calculate the #color(brown)(Gradien)t# of the #color(green)("Tangent")#
substituting #x=7# into the equation #f'(x)# : #f'(7)=(3*7^2)-(4*7)-6rArr113#
As we have the #color(brown)(Gradien)t# of the #color(green)("Tangent")# we can write that the equation of #color(green)("Tangent")# is
#y=113x+c#
where c is the #y#-intercept

Step 4: Find out what c is worth

substituting #x=7# & #y=216# into the above equation: #216=(113*7)+c# #:.c=-545#

Step 5: Find the Tangent equation

Now, substituting the value of #c=-545#
#color(red)(y=113x-545)# graph{(x^3-2x^2-6x+13-y)(113x-545-y)=0 [-10, 10, -5, 5]}
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Answer 2

The equation of the tangent line of f(x)=(x-1)^3+(x-2)(x-7) at x=7 is y = 6x - 29.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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