How do you find an equation of the tangent line to the curve at the given point #y=tan(4x^2) # and #x=sqrtpi#?
You can use the point and the slope in the pointslope form of a line
graph{(ytan(4x^2))(y8sqrt(pi)x+8pi)=0[1.2,2.2,3,3]}
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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:

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).

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).

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).

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.

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

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.

Use the pointslope 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.

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 36 may be quite complex, and it is recommended to use a calculator or software to evaluate the trigonometric functions accurately.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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