Examples of the tangent line approximation
Definition of the Derivative
- The derivative of a function represents the rate at which the function is changing at any given point.
- It measures the slope of the tangent line to the curve at that point.
- The derivative is denoted by
f'(x) or dy/dx.
Tangent Line Approximation
- The tangent line approximates the behavior of a function near a specific point.
- It is a straight line that touches the curve at only one point, which is the point of tangency.
- The tangent line can be used to estimate the value of the function at any given point.
Example 1: Finding the Tangent Line
- Find the equation of the tangent line to the curve
f(x) = x^2 at the point (2, 4).
- Solution:
- The derivative of
f(x) is 2x.
- At
x = 2, the slope of the tangent line is 2 x 2 = 4.
- Since the point
(2, 4) lies on the tangent line, the equation of the line is y - 4 = 4(x - 2).
Example 2: Approximating a Function Value
- Use the tangent line to approximate the value of
f(x) = sin(x) at x = π/6.
- Solution:
- The derivative of
f(x) is cos(x).
- At
x = π/6, the slope of the tangent line is cos(π/6) = √3/2.
- Since the point
(π/6, sin(π/6)) lies on the tangent line, the equation of the line is y - sin(π/6) = (√3/2)(x - π/6).
Example 3: Tangent Line as an Approximation
- Let
f(x) = ln(x) be the natural logarithm function.
- Use the tangent line at
x = 1 to approximate ln(1.1).
- Solution:
- The derivative of
f(x) is 1/x.
- At
x = 1, the slope of the tangent line is 1/1 = 1.
- Since the point
(1, ln(1)) lies on the tangent line, the equation of the line is y - ln(1) = 1(x - 1).
Example 4: Linear Approximation
- Let
f(x) = √x be the square root function.
- Use the tangent line at
x = 4 to approximate √3.
- Solution:
- The derivative of
f(x) is 1/(2√x).
- At
x = 4, the slope of the tangent line is 1/(2√4) = 1/4.
- Since the point
(4, √4) lies on the tangent line, the equation of the line is y - √4 = (1/4)(x - 4).
Example 5: Tangent Line to the Cubic Function
- Let
f(x) = x^3 be the cubic function.
- Find the equation of the tangent line at
x = -2.
- Solution:
- The derivative of
f(x) is 3x^2.
- At
x = -2, the slope of the tangent line is 3(-2)^2 = 12.
- Since the point
(-2, (-2)^3) lies on the tangent line, the equation of the line is y - (-2)^3 = 12(x - (-2)).
Example 6: Tangent Line to the Exponential Function
- Let
f(x) = e^x be the exponential function.
- Find the equation of the tangent line at
x = 1.
- Solution:
- The derivative of
f(x) is e^x.
- At
x = 1, the slope of the tangent line is e^1 = e.
- Since the point
(1, e) lies on the tangent line, the equation of the line is y - e^1 = e(x - 1).
Summary
- The derivative represents the slope of the tangent line to a curve at any given point.
- The tangent line approximation can be used to estimate function values.
- The equation of the tangent line is found using the slope and a point on the line.
- Tangent lines provide a good approximation when the function is locally linear.
Example 7: Tangent Line to a Rational Function
- Let
f(x) = 1/x be the rational function.
- Find the equation of the tangent line at
x = 2.
- Solution:
- The derivative of
f(x) is -1/x^2.
- At
x = 2, the slope of the tangent line is -1/(2^2) = -1/4.
- Since the point
(2, 1/2) lies on the tangent line, the equation of the line is y - 1/2 = (-1/4)(x - 2).
Example 8: Tangent Line to an Absolute Value Function
- Let
f(x) = |x| be the absolute value function.
- Find the equation of the tangent line at
x = -3.
- Solution:
- The derivative of
f(x) is -1 for x < 0 and 1 for x > 0.
- At
x = -3, the slope of the tangent line is -1.
- Since the point
(-3, |-3|) lies on the tangent line, the equation of the line is y - |-3| = -1(x - (-3)).
Example 9: Tangent Line to a Logarithmic Function
- Let
f(x) = log(x) be the logarithmic function.
- Find the equation of the tangent line at
x = 10.
- Solution:
- The derivative of
f(x) is 1/x.
- At
x = 10, the slope of the tangent line is 1/10.
- Since the point
(10, log(10)) lies on the tangent line, the equation of the line is y - log(10) = (1/10)(x - 10).
Example 10: Tangent Line to a Quadratic Function
- Let
f(x) = 2x^2 - 3x + 1 be a quadratic function.
- Find the equation of the tangent line at
x = 1.
- Solution:
- The derivative of
f(x) is 4x - 3.
- At
x = 1, the slope of the tangent line is 4(1) - 3 = 1.
- Since the point
(1, 2 - 3 + 1) lies on the tangent line, the equation of the line is y - (2 - 3 + 1) = 1(x - 1).
Example 11: Tangent Line to an Exponential Growth Function
- Let
f(x) = 2^x be an exponential growth function.
- Find the equation of the tangent line at
x = 0.
- Solution:
- The derivative of
f(x) is ln(2) * 2^x.
- At
x = 0, the slope of the tangent line is ln(2) * 2^0 = ln(2).
- Since the point
(0, 2^0) lies on the tangent line, the equation of the line is y - 2^0 = ln(2)(x - 0).
Example 12: Tangent Line to a Sine Function
- Let
f(x) = sin(x) be a sine function.
- Find the equation of the tangent line at
x = π/4.
- Solution:
- The derivative of
f(x) is cos(x).
- At
x = π/4, the slope of the tangent line is cos(π/4) = 1/√2.
- Since the point
(π/4, sin(π/4)) lies on the tangent line, the equation of the line is y - sin(π/4) = (1/√2)(x - π/4).
Tangent Line to an Inverse Function
- The tangent line to
f(x) at x = a is perpendicular to the tangent line to f^(-1)(x) at y = f(a).
- The slope of the tangent line to
f(x) at x = a is the reciprocal of the slope of the tangent line to f^(-1)(x) at y = f(a).
Tangent Line to a Parametric Curve
- For a parametric curve, the slope of the tangent line is given by
dy/dx, where y and x are expressed in terms of a third parameter t.
- The equation of the tangent line at a point
(x0, y0) is given by y - y0 = dy/dx(x - x0).
Tangent Line to a Polar Curve
- For a polar curve
r = f(θ), the slope of the tangent line at a point (r0, θ0) is given by (dy/dθ)/(dx/dθ).
- The equation of the tangent line at a point
(r0, θ0) is given by r - r0 = (dy/dθ)/(dx/dθ)(θ - θ0).
Conclusion: Tangent Line Approximation
- Tangent lines provide a good approximation to the behavior of a function at a specific point.
- They can be used to estimate function values and understand the local behavior of a curve.
- The equation of a tangent line is determined by its slope and a point on the line.
- Tangent lines can also be applied to inverse, parametric, and polar curves.
Examples of the tangent line approximation
Example 13: Tangent Line to a Piecewise Function
- Let
f(x) = x^2 for x > 0 and f(x) = 2x + 1 for x ≤ 0 be a piecewise function.
- Find the equation of the tangent line at
x = 1.
- Solution:
- The derivative of
f(x) is 2x for x > 0 and 2 for x ≤ 0.
- At
x = 1, the slope of the tangent line is 2(1) = 2.
- Since the point
(1, 1^2) lies on the tangent line, the equation of the line is y - 1^2 = 2(x - 1).
Example 14: Tangent Line to a Logarithmic Function
- Let
f(x) = ln(x) be the natural logarithm function.
- Find the equation of the tangent line at
x = e.
- Solution:
- The derivative of
f(x) is 1/x.
- At
x = e, the slope of the tangent line is 1/e.
- Since the point
(e, ln(e)) lies on the tangent line, the equation of the line is y - ln(e) = (1/e)(x - e).
Example 15: Tangent Line to a Trigonometric Function
- Let
f(x) = cos(x) be a cosine function.
- Find the equation of the tangent line at
x = π/3.
- Solution:
- The derivative of
f(x) is -sin(x).
- At
x = π/3, the slope of the tangent line is -sin(π/3) = -√(3)/2.
- Since the point
(π/3, cos(π/3)) lies on the tangent line, the equation of the line is y - cos(π/3) = (-√(3)/2)(x - π/3).
Example 16: Tangent Line to an Exponential Decay Function
- Let
f(x) = e^(-x) be an exponential decay function.
- Find the equation of the tangent line at
x = 0.
- Solution:
- The derivative of
f(x) is -e^(-x).
- At
x = 0, the slope of the tangent line is -e^0 = -1.
- Since the point
(0, e^0) lies on the tangent line, the equation of the line is y - e^0 = -1(x - 0).
Example 17: Tangent Line to a Power Function
- Let
f(x) = x^3 be a cubic function.
- Find the equation of the tangent line at
x = 2.
- Solution:
- The derivative of
f(x) is 3x^2.
- At
x = 2, the slope of the tangent line is 3(2)^2 = 12.
- Since the point
(2, (2)^3) lies on the tangent line, the equation of the line is y - (2)^3 = 12(x - 2).
Example 18: Tangent Line to a Rational Function
- Let
f(x) = (x - 1)/(x + 1) be a rational function.
- Find the equation of the tangent line at
x = -1.
- Solution:
- The derivative of
f(x) is ((x + 1) - (x - 1))/((x + 1)^2) = 2/(x + 1)^2.
- At
x = -1, the slope of the tangent line is 2/(-1 + 1)^2 = 2.
- Since the point
(-1, (-1 - 1)/(-1 + 1)) lies on the tangent line, the equation of the line is y - (-1 - 1)/(-1 + 1) = 2(x - (-1)).
Example 19: Tangent Line to an Absolute Value Function
- Let
f(x) = |x - 2| be the absolute value function.
- Find the equation of the tangent line at
x = 2.
- Solution:
- The derivative of
f(x) is 1 for x > 2 and -1 for x < 2.
- At
x = 2, the slope of the tangent line is 1.
- Since the point
(2, |2 - 2|) lies on the tangent line, the equation of the line is y - |2 - 2| = 1(x - 2).
Example 20: Tangent Line to a Logarithmic Function
- Let
f(x) = log(2x) be the logarithmic function.
- Find the equation of the tangent line at
x = 1.
- Solution:
- The derivative of
f(x) is 1/(2x).
- At
x = 1, the slope of the tangent line is 1/(2(1)) = 1/2.
- Since the point
(1, log(2(1))) lies on the tangent line, the equation of the line is y - log(2(1)) = (1/2)(x - 1).
Real-World Application: Tangent Line Approximation in Physics
- In physics, the tangent line approximation is commonly used when analyzing the motion of objects.
- The tangent line represents the instantaneous velocity of an object at a specific point in time.
- By finding the equation of the tangent line, we can estimate the position and velocity of the object at any given time.
- This approximation is particularly useful when dealing with non-linear motion or complex functions.
Real-World Application: Tangent Line Approximation in Economics
- In economics, the tangent line approximation is often used to estimate the marginal impact of changes in variables.
- By finding the equation of the tangent line to a production, cost, or revenue function, we can determine the rate of change at a specific level of output.
- This information helps businesses make decisions on pricing,