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And draw the tangent lines to the graph at points whose x-coordinates are -2 , 0, and 1.

And draw the tangent lines to the graph at points whose x-coordinates are -2 , 0, and-example-1
And draw the tangent lines to the graph at points whose x-coordinates are -2 , 0, and-example-1
And draw the tangent lines to the graph at points whose x-coordinates are -2 , 0, and-example-2
User Jislam
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1 Answer

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a)

To find the difference quotient we will do


(f(x+h)-f(x))/(h)

But f(x) = 4x, therefore f(x+h) = 4(x+h). Then


(f(x+h)-f(x))/(h)=(4(x+h)-4x)/(h)

Now we just simplify the expression


(4(x+h)-4x)/(h)=(4x+4h-4x)/(h)=(4h)/(h)=4

Therefore


(f(x+h)-f(x))/(h)=4

As expected, because the function f is a linear function, then it must be a constant.

b)

As we can see we don't have the term "h" in the difference quotient, then we can easily solve the limit by just repeating the result:


\lim _(h\rightarrow0)(f(x+h)-f(x))/(h)=\lim _(h\rightarrow0)4=4

Therefore


f^(\prime)(x)=4

See that it's a constant function, it means that the slope will always be the same, doesn't matter the value of x.

User Ty Le
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