Calculus: Average Rate of Change and Derivatives

Average Rate of Change

• The average rate of change of a function f(x) over an interval from x = a to x = b is calculated as: (f(b) - f(a))/(b - a)
• In this case, f(x) = √(2 - 2x + 3) and the interval is from x = 3 to x = 11

Derivative Definition

• The derivative of a function f(x) at a point x = a, denoted as f'(a), is defined as the limit of the difference quotient as h approaches 0: f'(a) = lim(h->0) [(f(a + h) - f(a))/h]

Particle Motion

• The position of a particle at time t is given by p(t) = (1/(t + 1), 1 + e^(2t))
• The speed of the particle at time t is the magnitude of the velocity vector, which is the derivative of the position vector: |p'(t)|
• The acceleration of the particle at time t is the derivative of the velocity vector: p''(t)

Concavity and Critical Points

• A function is concave up on an interval where its second derivative is positive, and concave down where its second derivative is negative.
• Critical points of a function are points where the derivative is either zero or undefined.
• A function is monotone increasing on an interval where its derivative is positive, and monotone decreasing where its derivative is negative.
• Local minimum points and local maximum points occur at critical points where the derivative changes sign.

Differentiation Rules

• The derivative of f(x) = (cos x)e^(x+1) + 2sin x is f'(x) = -sin(x)e^(x+1) + (cos x)e^(x+1) + 2cos(x)
• The derivative of f(x) = e^(ln(2x - x^3)) + (cos(3x))/(x^2 + 4) is f'(x) = (2 - 3x^2)e^(ln(2x - x^3)) - (3sin(3x)(x^2 + 4) - 2xcos(3x))/(x^2 + 4)^2
• The derivative of f(x) = (cos x)/(x^2 + 4) is f'(x) = (-sin(x)(x^2 + 4) - 2xcos(x))/(x^2 + 4)^2

Logarithmic Differentiation

• Logarithmic differentiation is a technique used to find the derivative of functions that are products, quotients, or powers of functions
• To use logarithmic differentiation, take the natural logarithm of both sides of the equation, simplify, and then differentiate implicitly

Tangent Line

• The equation of the tangent line to a curve at a point (a, f(a)) is given by y - f(a) = f'(a)(x - a)

Inverse Functions

• An inverse function reverses the input and output of the original function.
• The derivative of an inverse function at a point f(a) is given by (f^-1)'(f(a)) = 1/f'(a)