This is the first step into multivariable calculus. A partial derivative of a function of several variables (e.g., f(x, y)) is its derivative with respect to one of those variables, while holding all other variables constant.
Let z = f(x, y).
To calculate, treat 'y' as if it were a constant (like '5') and differentiate f with respect to 'x' normally.
To calculate, treat 'x' as if it were a constant and differentiate f with respect to 'y' normally.
Example: Let f(x, y) = x³ + x²y⁵ - 2y⁴
We can differentiate again to find second-order partial derivatives:
For most "well-behaved" functions (if f, fₓ, fᵧ, fₓᵧ, and fᵧₓ are all continuous), the order of differentiation does not matter.
∂²f/∂y∂x = ∂²f/∂x∂y (or fₓᵧ = fᵧₓ)
A function f(x, y) is a homogeneous function of degree n if, for any constant t, it satisfies the following property:
f(tx, ty) = tⁿ * f(x, y)
If every term in a polynomial function has the same total degree, the function is homogeneous. The total degree of a term like xᵃyᵇ is a + b.
Example: f(x, y) = 3x⁵ + 2x²y³ - y⁵
Since all terms have degree 5, this is a homogeneous function of degree 5.
Example (Rational Function): f(x, y) = (x³ + y³) / (x - y)
f(tx, ty) = ((tx)³ + (ty)³) / (tx - ty) = (t³x³ + t³y³) / (tx - ty)
= [t³(x³ + y³)] / [t(x - y)] = t² * [(x³ + y³) / (x - y)]
= t² * f(x, y)
This is a homogeneous function of degree 2.
This theorem provides a beautiful relationship between a homogeneous function and its partial derivatives.
If f(x, y) is a homogeneous function of degree n, then: x * (∂f/∂x) + y * (∂f/∂y) = n * f
Example: Verify Euler's Theorem for f(x, y) = 3x⁵ + 2x²y³ - y⁵ (n = 5)
If f(x, y) is a homogeneous function of degree n, then: x²(∂²f/∂x²) + 2xy(∂²f/∂x∂y) + y²(∂²f/∂y²) = n(n-1) * f
An asymptote is a line that a curve approaches (gets infinitely close to) as the x or y coordinates approach infinity.
These occur where the function "blows up" to ±∞. For a rational function f(x) = P(x) / Q(x), vertical asymptotes are found at the x-values 'a' that make the denominator zero (and the numerator non-zero).
Example: y = 1 / (x - 2). The denominator is zero at x = 2. So, x = 2 is a vertical asymptote.
These describe the behavior of f(x) as x → ∞ and x → -∞. y = L is a horizontal asymptote if lim (x → ∞) f(x) = L or lim (x → -∞) f(x) = L.
For a rational function f(x) = (axⁿ + ...) / (bxᵐ + ...):
These occur in rational functions when the degree of the numerator is exactly one more than the degree of the denominator.
The equation of the slant asymptote (y = mx + c) is the quotient obtained by performing polynomial long division.
Example: f(x) = (x² + 1) / (x - 1)
This topic uses the second derivative, f''(x), to describe the "bend" of a curve.
A point of inflection is a point on the curve where the concavity changes (from CU to CD, or from CD to CU).
Example: f(x) = x³
This is a capstone topic that combines all the concepts from the course. To trace a graph of y = f(x), you should follow a systematic procedure.
Find the set of all possible x-values (domain) and y-values (range). Look for:
- Division by zero (for rational functions).
- Square roots of negative numbers.
Y-intercept: Set x = 0 and solve for y. (Point is (0, y)).
X-intercept(s): Set y = 0 and solve for x. (Points are (x, 0)).
Y-axis Symmetry (Even Function): Is f(-x) = f(x)? (e.g., y = x², y = cos x)
Origin Symmetry (Odd Function): Is f(-x) = -f(x)? (e.g., y = x³, y = sin x)
If f(x,y)=0 is the equation, check if replacing x with -x (y-axis) or y with -y (x-axis) or both (origin) leaves the equation unchanged.
Find all Vertical, Horizontal, and Oblique asymptotes as described in the section above.
Find f'(x).
Find critical points by solving f'(x) = 0 or where f'(x) is undefined.
Create a sign chart for f'(x) to find intervals of increasing (f' > 0) and decreasing (f' < 0).
Use the First Derivative Test to classify critical points as local maxima or local minima.
Find f''(x).
Find potential inflection points by solving f''(x) = 0 or where f''(x) is undefined.
Create a sign chart for f''(x) to find intervals of concave up (f'' > 0) and concave down (f'' < 0).
Identify points of inflection (where concavity changes).
Draw the axes and mark the asymptotes (as dashed lines).
Plot all key points: intercepts, local max/mins, and inflection points.
Connect the dots, making sure your curve follows the information from your sign charts (increasing/decreasing and concavity) and approaches the asymptotes correctly.