How do you show that the point #(p,4p^2)# lies on the curve #y=4x^2# for all real values of p and then find the equation of the tangent to #y=4x^2# at #(p,4p^2)#?

Answer 1
You can substitute the coordinate #x=p# of your point into the function to get: #y=4p^2# so your point does lie on the curve represented by #y=4x^2#.
Your function is a quadratic which can accept all real values of #x# so that #p# can be every real value.
Deriving your function you get the slope of the tangent to the curve at a generic #x#: #y'=8x#
The slope in #x=p# is: #y'(p)=8p#
The equation of the line through your point (of coordinates #(x_0,y_0)#) and slope #m=8p# is: #y-y_0=m(x-x_0)# #y-4p^2=8p(x-p)# and finally: #y=8px-4p^2#
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Answer 2

To show that the point (p,4p^2) lies on the curve y=4x^2, we substitute the x-coordinate p into the equation and check if it satisfies the equation.

Substituting p into the equation y=4x^2, we get: y = 4p^2

Since the y-coordinate of the point (p,4p^2) is equal to 4p^2, it satisfies the equation y=4x^2. Therefore, the point (p,4p^2) lies on the curve y=4x^2 for all real values of p.

To find the equation of the tangent to y=4x^2 at (p,4p^2), we need to find the slope of the tangent at that point. The slope of the tangent can be found by taking the derivative of the equation y=4x^2 with respect to x.

Differentiating y=4x^2 with respect to x, we get: dy/dx = 8x

Substituting the x-coordinate p into the derivative, we get the slope of the tangent at (p,4p^2): dy/dx = 8p

Now, we have the slope of the tangent at (p,4p^2) as 8p. To find the equation of the tangent, we use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the point (p,4p^2) and m is the slope 8p.

Substituting the values into the point-slope form, we get the equation of the tangent: y - 4p^2 = 8p(x - p)

Simplifying the equation, we can expand the right side: y - 4p^2 = 8px - 8p^2

Finally, rearranging the equation, we get the equation of the tangent to y=4x^2 at (p,4p^2): y = 8px - 8p^2 + 4p^2

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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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