Intro to Calculus

Fundamentals of Gradient and Continuity

  • Continuous vs. Discontinuous Functions: A function is continuous if its graph is smooth and unbroken without gaps or breaks.
  • Differentiability at a Point: A function y=f(x)y = f(x) is differentiable at x=ax = a if its graph is both continuous and smooth at that point. Linear, quadratic, and cubic polynomials are differentiable at all points.
  • Angle of Inclination: The relationship between the angle of inclination θ\theta (with the positive xx-axis) and the gradient mm of a line or tangent is established as:
    tan⁡(θ)=m\tan(\theta) = m
  • Gradients of Specific Lines:
    • Horizontal line: m=0m = 0
    • Vertical line: m=undefinedm = \text{undefined}
    • Increasing line: m>0m > 0
    • Decreasing line: m<0m < 0

Rates of Change and Limits

  • Average Rate of Change: Represented by the gradient of a secant (a line passing through two points on a curve). It is calculated using the difference quotient:
    f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h}
  • Instantaneous Rate of Change: Represented by the gradient of the tangent at a specific point. This is found by taking the limit of the difference quotient as the distance between points hh approaches zero.
  • First Principles Definition: The derivative f′(x)f'(x) is defined as:
    f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
  • Derivative Notation: The derivative can be denoted as dydx\frac{dy}{dx}, f′(x)f'(x), or y′y' where y=f(x)y = f(x).

Rules for Differentiation

  • Power Rule: For all real values of nn, ddx(xn)=nxn−1\frac{d}{dx}(x^n) = nx^{n-1}.
  • Constant Multiple Rule: ddx(kf(x))=kf′(x)\frac{d}{dx}(kf(x)) = kf'(x).
  • Sum and Difference Rule: ddx(f(x)±g(x))=f′(x)±g′(x)\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x).
  • Product Rule: If h(x)=f(x)g(x)h(x) = f(x)g(x), then:
    h′(x)=f(x)g′(x)+f′(x)g(x)h'(x) = f(x)g'(x) + f'(x)g(x)
  • Quotient Rule: If h(x)=f(x)g(x)h(x) = \frac{f(x)}{g(x)}, then:
    h′(x)=g(x)f′(x)−f(x)g′(x)[g(x)]2h'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}
  • Chain Rule: For composite functions y=f(g(x))y = f(g(x)):
    dydx=dydu×dudx\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

Applications of the Derivative

  • Function Behavior:
    • Stationary Point: A point where the tangent is horizontal and f′(x)=0f'(x) = 0. This includes maximums, minimums, or horizontal points of inflection.
    • Increasing Function: Where the gradient f′(x)>0f'(x) > 0.
    • Decreasing Function: Where the gradient f′(x)<0f'(x) < 0.
  • Tangents and Normals:
    • The tangent is a straight line that touches the curve at 1 point, sharing its gradient.
    • The normal is a line perpendicular to the tangent. Their gradients m1m_1 and m2m_2 satisfy m1×m2=−1m_1 \times m_2 = -1 or m2=−1m1m_2 = -\frac{1}{m_1}.
  • Kinematics:
    • Velocity: The rate of change of displacement with respect to time.
    • Acceleration: The rate of change of velocity with respect to time.

Mathematical Terminology

  • Differentiation: The process of finding the gradient function.
  • Displacement: The distance and direction of an object in relation to the origin.
  • Limit: The value a function approaches as the independent variable approaches a specific value.
  • Turning Point: A maximum or minimum point on a curve where the curve turns around.