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In the study of Differential Calculus, in general, the first application of limit that we learn allows us to obtain the equation of the tangent line to the graph of a given function f at a point P(x,y). This concept known as Derivative has numerous applications that allow us to measure rates of variation, such as speed, acceleration, growth, decrease, etc. Thus, consider f(x) = x2 a real function defined on every line R, determine the equation of the tangent line to this curve at the point P(2,4).

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Answer to a math question In the study of Differential Calculus, in general, the first application of limit that we learn allows us to obtain the equation of the tangent line to the graph of a given function f at a point P(x,y). This concept known as Derivative has numerous applications that allow us to measure rates of variation, such as speed, acceleration, growth, decrease, etc. Thus, consider f(x) = x2 a real function defined on every line R, determine the equation of the tangent line to this curve at the point P(2,4).

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Fred
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118 Answers
1. Given the function f(x) = x^2 and a point P(2, 4), we need to find the derivative of f(x) using the limit definition:

f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h}

2. Substitute f(x) = x^2:

f'(x) = \lim_{{h \to 0}} \frac{{(x + h)^2 - x^2}}{h}

3. Simplify the expression inside the limit:

f'(x) = \lim_{{h \to 0}} \frac{{x^2 + 2xh + h^2 - x^2}}{h}

f'(x) = \lim_{{h \to 0}} \frac{{2xh + h^2}}{h}

4. Factor out h:

f'(x) = \lim_{{h \to 0}} \frac{{h(2x + h)}}{h}

f'(x) = \lim_{{h \to 0}} (2x + h)

5. As h approaches 0:

f'(x) = 2x

6. Find the slope at x = 2:

f'(2) = 2(2) = 4

7. Use the point-slope form of the equation of a line with slope m = 4 and point (2, 4):

y - 4 = 4(x - 2)

8. Simplify to the final equation:

y - 4 = 4x - 8

y = 4x - 4

Answer: y = 4x - 4

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