What is the equation of the normal line of #f(x)=e^x-x^3# at #x=0#?

Answer 1

In slope intercept form:

#y = -x+1#

Given #f(x) = e^x-x^3#
Then #f'(x) = e^x-3x^2#
#f(0) = e^0 - 0 = 1#
So the graph of the function passes through #(0, 1)#
#f'(0) = e^0 - 0 = 1#
So the slope of the tangent at #(0, 1)# is #1#.
If the slope of the tangent is #m# then the slope of the normal is #-1/m#. So in our example the slope of the normal is #-1/1 = -1#.
The equation of a line of slope #m# passing through a point #(x_0, y_0)# can be written as:
#(y - y_0) = m(x - x_0)#
In our case we have #(x_0, y_0) = (0, 1)# and #m = -1#, so:
#(y - 1) = -1(x - 0) = -x#
That is #y = -x+1# in slope intercept form.

graph{(e^x-x^3-y)(x+y-1) = 0 [-10, 10, -5, 5]}

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Answer 2

The equation of the normal line of f(x)=e^x-x^3 at x=0 is y = -3x + 1.

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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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