How do you find the equation of a line tangent to the function #y=x2x^2+3# at x=2?
The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point.
so If
# dy/dx = 14x #
When
and
So the tangent passes through
# \ \ \ \ \ y(3) = 7(x2) #
# :. y+3 = 7x+14#
# :. \ \ \ \ \ \ \ y = 7x+11 #
We can confirm this solution is correct graphically:
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To find the equation of a line tangent to a function at a specific point, you need to follow these steps:
 Find the derivative of the function.
 Evaluate the derivative at the given xvalue to find the slope of the tangent line.
 Use the pointslope form of a line, y  y₁ = m(x  x₁), where (x₁, y₁) is the given point and m is the slope, to write the equation of the tangent line.
Let's apply these steps to the given function y = x  2x^2 + 3 at x = 2:

Find the derivative of the function: y' = 1  4x

Evaluate the derivative at x = 2: y'(2) = 1  4(2) = 7

Use the pointslope form with the point (2, y(2)): y  y₁ = m(x  x₁) y  y(2) = 7(x  2)
Simplifying the equation: y  (2  2(2)^2 + 3) = 7(x  2) y  (5) = 7(x  2) y + 5 = 7x + 14
The equation of the line tangent to the function y = x  2x^2 + 3 at x = 2 is: y = 7x + 9
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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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