EngMath Derivatives and Rates of Change.
Derivatives and Rates of Change
Introduction
The course MAT 115 explores derivatives and their significance in calculating rates of change and determining tangent lines to curves.
Previous material has established a rigorous understanding of limits, which is foundational for understanding derivatives.
Tangents and the Derivative
Definition of a Tangent Line
Let a curve be defined by the equation y = f(x).
The goal is to find the tangent line at a specific point P(a, f(a)).
A tangent line touches the curve at point P, indicating its direction at that point.
Secant Line
To find the equation of the tangent line, consider another point Q(x, f(x)), which is very close to P but not equal to it.
The line connecting P and Q is called a secant line.
The slope of the secant line, denoted as mPQ, is given by:
m_{PQ} = \frac{f(x) - f(a)}{x - a}As point Q approaches point P (i.e., as x approaches a), the slope of the secant line approaches a specific value m.
This value also represents the slope of the tangent line at point P.
Tangent Line Equation
The equation of the tangent line l at P with slope m can be defined as:
The secant line's slope as h approaches 0 is given by:
m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
This definition leads to the derivative of the function.
Definition of the Derivative
Definition
The derivative of a function f at the point a is defined as:
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}Alternatively, it can be expressed as:
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
Implications
The derivative represents the instantaneous rate of change of the function at a specific point.
It shows how f changes as x changes near the point a.
Examples
Example 1.1
Find the equation of the tangent line to the parabola y = x² at P(1, 1).
Example 1.2
Calculate the tangent line to the hyperbola y = 3/x at the point (3, 1).
Rates of Change in Sciences
Limits of the form:
\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
are prevalent in sciences for calculating rates of change (e.g., reaction rates in chemistry, marginal cost in economics).
Derivative as a Function
Expanding the Definition
The perspective of the derivative can be broadened by letting a be variable, thus:
f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}This expression defines a new function, the derivative function of f.
The geometric interpretation is that f'(x) gives the slope of the tangent line at point (x, f(x)).
The domain of f' may differ from that of f, as limits defining derivatives may not exist at all points.
Example Derivatives
Example 1.5: Determine the derivative of f(x) = 1/(1 - 4x).
Example 1.6:
(a) Find f'(x) for f(x) = x³ − x.
(b) Compare graphs of f and its derivative.
Example 1.7: For f(x) = √x, find the derivative and state the domain of f'.
Example 1.8: Derivative for f(x) = 1 - x² + x.
Notations for Derivatives
Common Notations
Leibniz Notation:
\frac{dy}{dx} or \frac{df}{dx} emphasizes the derivative’s rate of change of y with respect to x.
Lagrange Notation:
f'(x) is compact and standard in calculus.
Newton Notation:
\dot{y} is used mainly in physics for time derivatives.
Euler Notation:
D f(x) signifies the differentiation operator.
Higher Derivatives:
Second derivative: f''(x) or \frac{d^2y}{dx^2}.
nth derivative: f^{(n)}(x) or \frac{d^ny}{dx^n}.
Evaluating Derivatives at Points
For evaluating a derivative at a specific point x = a, use:
\frac{dy}{dx} \Big|_{x=a}Example: For f(x) = x³ − 4x, finding the derivative at x = 2 gives:
\frac{dy}{dx} \Big|_{x=2} = 3(2)^2 − 4 = 12 − 4 = 8.
Differentiability
Definition
A function f is differentiable at a if f'(a) exists.
A function is differentiable over an interval (a, b) if differentiable at every point within it.
Examples
Example 1.9: Determine if f(x) = {√2 − x, x ≤ 1; 2x - 1, x > 1} is differentiable at x = 1.
Example 1.10: Analyze the differentiability of the function f(x) = |x|.
Example 1.11: For function f(x) = {√2 − x², x ≤ 1; ax + b, x > 1}, find values of a and b for differentiability at x = 1.
Theorems and Proofs
Theorem 1.1
If f is differentiable at a, then f is continuous at a.
Proof involves showing that the limit of f(x) as x approaches a equals f(a).
Ways to Fail Differentiability
Functions fail to be differentiable under these conditions:
Sharp Corners or Cusps:
Example: f(x) = |x| at x=0 is continuous but not differentiable.
Vertical Tangents:
Example: f(x) = x^(1/3) at x=0 has an infinite slope, hence non-differentiable.
Discontinuities:
Any discontinuity results in non-differentiability (e.g., jump discontinuity).
Higher Derivatives
Concept and Definition
If f is differentiable, its derivative f' is also a function, with a derivative of its own called the second derivative, written as f''(x).
Higher derivatives (third, fourth, etc.) represent rates of change of prior derivatives:
e.g., f^{(3)}(x) is the derivative of f''(x).
Physical Interpretations
Acceleration:
If s = s(t) is the position function, its first derivative gives the velocity v(t) and its second derivative provides the acceleration a(t):
a(t) = v′(t) = s''(t)\text{ or } a = \frac{dv}{dt} = \frac{d^2s}{dt^2}.
Acceleration indicates the change in velocity, vital in physics applications (e.g., cars speeding up/slowing down).
Examples of Higher Derivatives
Example 1.12: Given f(x) = x³ − x, find and interpret f''(x).
Differentiation of Polynomial and Exponential Functions
Constant Functions
For a constant function f(x) = c:
Its derivative is zero:
f'(x) = 0.Proof: \lim_{h \to 0} \frac{c - c}{h} = 0.
Power Functions
For functions of the form f(x) = x^n (n as positive integer), the derivatives follow a pattern:
Examples:
\frac{d}{dx}(x^1) = 1
\frac{d}{dx}(x^2) = 2x
\frac{d}{dx}(x^3) = 3x^2
Power Rule (Theorem 1.2): If n is a positive integer, then: \frac{d}{dx}(x^n) = nx^{n-1}.
Examples of this rule:
If f(x) = x⁶, then f′(x) = 6x⁵.
Power Functions with Negative Exponents
For negative integer exponents:
The derivative holds as:
\frac{d}{dx}(x^{-1}) = -1/x^2.
Power Functions with Fractional Exponents
For fractional exponents:
\frac{d}{dx}(√x) = \frac{1}{2√x}.
General Power Rule (Theorem 1.3):
If n is any real number, then:
\frac{d}{dx}(x^n) = nx^{n-1}.
Examples of Derivatives
Example 1.14: Differentiate f(x) = 1/x² and y = √3x².
Tangent and Normal Lines
Concepts
The tangent line to a curve represents the best linear approximation at a point.
The normal line is perpendicular to the tangent line, important in describing perpendicular directions (as in physics).
Example 1.15
Find equations of the tangent line and normal line for y = x√x at the point (1, 1).
New Derivatives from Old Functions
The Constant Multiple Rule (Theorem 1.4)
If c is a constant and f is differentiable, then: \frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x).
Example: \frac{d}{dx}(3x^4) = 3 \cdot 4x^3 = 12x^3.
The Sum and Difference Rules (Theorem 1.5)
If f and g are both differentiable, then:
\frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)
\frac{d}{dx}[f(x) - g(x)] = \frac{d}{dx}f(x) - \frac{d}{dx}g(x).
This can be extended to any number of functions.
Differentiating Polynomials
Examples: Using the product, sum, and power rules combine to differentiate any polynomial functions smoothly.
Differentiation of Exponential Functions
Understanding the Natural Exponential Function
To compute the derivative of f(x) = b^x, follow:
f′(x) = \lim{h \to 0} \frac{b^{x+h} - b^x}{h} = \lim{h \to 0} \frac{b^x(b^h - 1)}{h}.
The value of this limit at h=0 is related to its derivative at zero, leading to proportionality of the slope to the function height.
Derivative of Natural Exponential Function (Theorem 1.6)
For the special case of the base e (Euler's number), the derivative simplifies:
\frac{d}{dx}(e^x) = e^x.
This means the function is its own derivative, relating slope directly to y-value at any point.
Examples of Exponential Functions
Example 1.18: If f(x) = e^x − x, find f′ and f′′.
The Product and Quotient Rules
The Product Rule (Theorem 1.7)
To differentiate the product of two functions:
If u = f(x) and v = g(x), then:
\frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx}.
Illustrative example: If f(x) = x and g(x) = x², find derivatives.
The Quotient Rule (Theorem 1.8)
For the quotient of two differentiable functions:
The derivative formula is:
\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x) \frac{d}{dx}f(x) - f(x) \frac{d}{dx}g(x)}{[g(x)]^2}.
The differentiation process requires attention to both numerator and denominator slopes.
Examples Using Product and Quotient Rules
Example 1.24: Differentiate y = (x²+x−2)/(x³+6).
Example 1.25: Analyze function y = x/(1−x) for derivatives and tangent lines.
Example 1.26: Find tangent line for curve y = (e^x)/(1+x²).
Summary of Differentiation Formulas
Deriving Formula | Result |
|---|---|
Constant | \frac{d}{dx}(c) = 0 |
Power Function | \frac{d}{dx}(x^n) = nx^{n−1} |
Constant Multiple | \frac{d}{dx}[cf(x)] = c\frac{d}{dx}f(x) |
Sum | \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) |
Product | \frac{d}{dx}[fg] = f'g + fg' |
Quotient | \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{gf' - fg'}{g^2} |
Concluding Examples
Example 1.27: Compute limits as derivatives for specific cases.
Example 1.28: Determine tangent lines that are parallel to specific equations of lines in Cartesian space.
Example 1.29: Examine various differentiable functions to compute their derivatives and tangent properties.