How do you find an equation of the tangent line to the curve at the given point #y=tan(4x^2) # and #x=sqrtpi#?

Answer 1

#y=8sqrt(pi)x-8pi#

When #x=sqrt(pi)#, then
#y=tan(4(sqrt(pi))^2)#
#y=tan(4pi)#
#y=0#
So we have the point #(sqrt(pi),0)#. The slope, #m#, is found by the derivative at this point.
The derivative of #tan(u)# is #sec^2(u)(du)/dx#
#d/dx tan(4x^2)=sec^2(4x^2)xx8x#
When #x=sqrt(pi)#, the slope is
#m=sec^2(4(sqrt(pi))^2)xx8(sqrt(pi))#
#m=(8sqrt(pi))/cos^2(4pi)#
#m=(8sqrt(pi))/1^2~~14.18#

You can use the point and the slope in the point-slope form of a line

#y-y_1=m(x-x_1)#
#y-0=8sqrt(pi)(x-sqrt(pi))#
#y=8sqrt(pi)x-8pi#
Here is a graph of the tangent function with the tangent line passing through the point #(sqrt(pi),0)#

graph{(y-tan(4x^2))(y-8sqrt(pi)x+8pi)=0[1.2,2.2,-3,3]}

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Answer 2

To find the equation of the tangent line to the curve y = tan(4x^2) at the point (x, y) where x = √π, you can follow these steps:

  1. Find the derivative of the function y = tan(4x^2) using the chain rule. The derivative is given by dy/dx = 8x * sec^2(4x^2).

  2. Substitute x = √π into the derivative to find the slope of the tangent line at that point. This gives us dy/dx = 8√π * sec^2(4(√π)^2).

  3. Evaluate the value of sec^2(4(√π)^2) using the trigonometric identity sec^2(x) = 1 + tan^2(x). In this case, sec^2(4(√π)^2) = 1 + tan^2(4(√π)^2).

  4. Calculate the value of tan(4(√π)^2) and substitute it into the equation from step 3. This gives us sec^2(4(√π)^2) = 1 + tan^2(4(√π)^2) = 1 + tan^2(t), where t = 4(√π)^2.

  5. Simplify the equation from step 4 to find sec^2(t).

  6. Substitute the value of sec^2(t) into the derivative dy/dx = 8√π * sec^2(4(√π)^2) to find the slope of the tangent line at x = √π.

  7. Use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point (x, y) and m is the slope found in step 6, to write the equation of the tangent line.

  8. Substitute the values of x, y, and m into the equation from step 7 to obtain the final equation of the tangent line.

Please note that the calculations involved in steps 3-6 may be quite complex, and it is recommended to use a calculator or software to evaluate the trigonometric functions accurately.

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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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