Overview
This lecture focused on the definition and interpretation of derivatives as slopes of tangent lines, the process of finding derivatives using limits, and a variety of examples applying these concepts to different types of functions. It also covered how to find tangent lines that are parallel or perpendicular to a given line, and how to find lines tangent to two different curves.
Tangent and Secant Lines
- The tangent line touches a curve at exactly one point; "tangent" means "to touch."
- The secant line crosses the curve at two points, providing two points to work with.
- As the two points defining a secant line get closer together (as B approaches A), the secant line approaches the tangent line. This process is described using limits.
- The tangent line gives the best linear approximation to the curve at a point, while the secant line gives an average rate of change between two points.
The Derivative as a Limit
- The derivative at a point A, denoted F'(A), represents the instantaneous rate of change or the slope of the tangent line at A.
- Limit definition:
F'(A) = lim_{H→0} [F(A+H) - F(A)] / H
- Alternate form:
F'(A) = lim_{B→A} [F(B) - F(A)] / (B - A)
- The average rate of change (secant slope) is [F(B) - F(A)] / (B - A). As B approaches A, this becomes the instantaneous rate of change (the derivative).
- The derivative can be interpreted as how fast the function is changing at a specific point, the slope of the tangent line, or the instantaneous rate of change.
Calculating Derivatives (Examples)
- For a quadratic function F(X) = 5X² + 7X + 2:
- F'(A) = lim_{H→0} [F(A+H) - F(A)] / H
- Plug in A+H everywhere X appears, expand, combine like terms, and simplify.
- Result: F'(A) = 10A + 7
- Note: Adding a constant (like "+2") to a function does not affect its derivative, since it only shifts the graph vertically and does not change the slope.
- For F(X) = |X| (absolute value function):
- At X = 0, the derivative does not exist because the left-hand and right-hand limits are different (from the right, the limit is 1; from the left, it is -1).
- The graph has a sharp corner at X = 0, so there is no single tangent line at that point.
- For F(X) = sin(4X) at X = 0:
- F(0) = 0; F'(0) = 4 (using the limit and the identity for sin(kH)/kH as H→0).
- The tangent line at X = 0 is Y = 4X.
- For F(X) = 1/√X at X = 4:
- F(4) = 1/2
- F'(4) = -1/16 (using the limit definition and algebraic manipulation, including rationalizing the numerator).
- The tangent line at X = 4 is Y - 1/2 = (-1/16)(X - 4)._
Tangent Line Formula & Approximation
- The point-slope form of the tangent line at X = A:
Y - F(A) = F'(A)(X - A)
- Alternate form:
Y = F(A) + F'(A)(X - A)
- Slope-intercept form:
Y = F'(A)X + [F(A) - A·F'(A)]
- The tangent line provides the best linear approximation to the function near X = A:
F(X) ≈ F(A) + F'(A)(X - A) when X is close to A.
Working with Tangent Lines
- The slope of the tangent line at a point is the value of the derivative at that point.
- To find the tangent line to a function at a given point:
- Find the value of the function at that point (F(A)).
- Find the value of the derivative at that point (F'(A)).
- Use the point-slope form to write the equation of the tangent line.
- If given a tangent line (e.g., Y = 3X - 7) to a function at a specific X value (e.g., X = 4):
- The slope of the tangent line is the value of the derivative at that point (F'(4) = 3).
- The value of the function at that point is found by plugging X = 4 into the tangent line (F(4) = 3·4 - 7 = 5).
- You can translate between information about the tangent line and information about the function and its derivative.
Tangent Lines and Parallel/Perpendicular Lines
- For Y = X², the derivative at X = A is 2A.
- To find tangent lines parallel to Y = X (slope 1):
- Set 2A = 1 ⇒ A = 1/2.
- The point is (1/2, 1/4), and the tangent line is Y = X - 1/4.
- To find tangent lines perpendicular to Y = X (slope -1):
- Set 2A = -1 ⇒ A = -1/2.
- The point is (-1/2, 1/4), and the tangent line is Y = -X - 1/4.
- Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.
Tangent to Two Curves
- To find a line tangent to both F(X) = 2X² + 4X + 2 and G(X) = -X² + 2X - 1:
- Let the tangent points be A on F and B on G.
- Set the slopes equal: F'(A) = G'(B).
- F'(A) = 4A + 4; G'(B) = -2B + 2.
- Set 4A + 4 = -2B + 2.
- Set the Y-intercepts of the tangent lines equal:
- For F: Y-intercept = F(A) - A·F'(A) = 2A² + 4A + 2 - A(4A + 4) = -2A² + 2.
- For G: Y-intercept = G(B) - B·G'(B) = -B² + 2B - 1 - B(-2B + 2) = B² - 1.
- Set -2A² + 2 = B² - 1.
- Solve the system for A and B.
- Substitute B = -1 - 2A into the second equation and solve for A.
- The solutions are A = -1 (corresponds to Y = 0) and A = 1/3 (corresponds to Y - 32/9 = (16/3)(X - 1/3)).
- The line Y = 0 is tangent to both curves at different points; the other tangent line is found by solving the system.
Key Terms & Definitions
- Tangent Line: A line that touches a curve at only one point and has the same slope as the curve at that point.
- Secant Line: A line that crosses a curve at two points, used to calculate the average rate of change.
- Derivative (F'(A)): The instantaneous rate of change of a function at point A; the slope of the tangent line at A.
- Limit: The value a function approaches as the input approaches a specified point; fundamental to the definition of the derivative.
- Instantaneous Rate of Change: The rate at which a function is changing at a specific point, given by the derivative.
- Average Rate of Change: The change in the function's output divided by the change in input between two points.
Action Items / Next Steps
- Practice applying the limit definition of the derivative to various types of functions, including polynomials, absolute value, trigonometric, and rational functions.
- Work on problems involving finding tangent lines that are parallel or perpendicular to a given line.
- Review how to translate between the equation of a tangent line and information about the function and its derivative at a point.
- Prepare for homework and quizzes that may require moving between tangent line equations and function/derivative data.