Calculus Basics

Calculus

Branches of Calculus

• Differential Calculus: deals with the study of rates of change and slopes of curves
• Integral Calculus: deals with the study of accumulation of quantities

Limits

• A limit represents the behavior of a function as the input (or x-value) approaches a specific value
• Notation: lim x→a f(x) = L
• Properties:
  • Sum rule: lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • Product rule: lim x→a [f(x) * g(x)] = lim x→a f(x) * lim x→a g(x)

Derivatives

• A derivative represents the rate of change of a function with respect to its input
• Notation: f'(x) or (d/dx)f(x)
• Rules:
  • Power rule: if f(x) = x^n, then f'(x) = nx^(n-1)
  • Product rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Chain rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Applications of Derivatives

• Finding maxima and minima: used to find the highest and lowest points of a function
• Finding the rate of change: used to find the rate at which a quantity changes with respect to another quantity
• Optimization problems: used to find the maximum or minimum value of a function subject to certain constraints

Integrals

• A definite integral represents the area between a curve and the x-axis over a specific interval
• Notation: ∫[a,b] f(x) dx
• Rules:
  • Constant multiple rule: ∫[a,b] k * f(x) dx = k * ∫[a,b] f(x) dx
  • Sum rule: ∫[a,b] [f(x) + g(x)] dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx

Applications of Integrals

• Area between curves: used to find the area between two curves
• Volume of solids: used to find the volume of solids with a circular base
• Work and energy: used to find the work done by a force and the energy of an object

微积分

微积分分支

• 微分积分: 研究变化率和曲线斜率
• 积分积分: 研究数量的累积

极限

• 极限表示函数在输入（或x值）接近特定值时的行为
• 记号：lim x→a f(x) = L
• 属性：
  • 和规则：lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • 积规则：lim x→a [f(x) * g(x)] = lim x→a f(x) * lim x→a g(x)

导数

• 导数表示函数相对于其输入的变化率
• 记号：f'(x) 或 (d/dx)f(x)
• 规则：
  • 幂规则：如果 f(x) = x^n，则 f'(x) = nx^(n-1)
  • 积规则：如果 f(x) = u(x)v(x)，则 f'(x) = u'(x)v(x) + u(x)v'(x)
  • 链式规则：如果 f(x) = g(h(x))，则 f'(x) = g'(h(x)) * h'(x)

导数应用

• 寻找最大值和最小值：用于寻找函数的最高和最低点
• 寻找变化率：用于寻找某一量相对于另一量的变化率
• 优化问题：用于寻找函数的最大或最小值，subject to certain constraints

积分

• 定积分表示曲线和x轴之间的面积， 在特定区间内
• 记号：∫[a,b] f(x) dx
• 规则：
  • 常数乘积规则：∫[a,b] k * f(x) dx = k * ∫[a,b] f(x) dx
  • 和规则：∫[a,b] [f(x) + g(x)] dx = ∫[a,b] f(x) dx + ∫[a,b] g(x) dx

积分应用

• 曲线之间的面积：用于寻找两曲线之间的面积
• 固体的体积：用于寻找圆基底固体的体积
• 工作和能量：用于寻找力所做的工作和物体的能量