Chapter 6: Calculus

Calculus describes change and accumulation, and it underlies all of physics and engineering (Stewart, 2016).

Limits and continuity

means gets arbitrarily close to as approaches . A function is continuous at if the limit exists and equals . Useful limits: (radians) and .

Differentiation

The derivative is , the gradient of the tangent. Rules: product ; quotient ; chain .

Also: implicit differentiation (differentiate both sides with respect to , treating as a function of ), parametric , and logarithmic differentiation for products and powers.

Worked example. . By the product rule, .

Applications of differentiation

Tangents and normals: the tangent at has gradient ; the normal has gradient .

Stationary points: . If it is a minimum; if a maximum; if test further (for example by sign changes of ).

Curve sketching: intercepts, asymptotes, stationary points, and behaviour as .

Rates of change: related rates use the chain rule.

Optimization. Example: a farmer has 100 m of fence to enclose a rectangle. If one side is , the other is and the area is . gives , a maximum since . The best shape is a square with area .

Integration

Integration reverses differentiation: . Standard integrals include (), , , , , and .

Techniques: substitution (choose so that appears); integration by parts ; partial fractions; trigonometric identities (for example ).

Worked example. : take , , so , . Then .

Definite integrals, area, and volume

The fundamental theorem of calculus: where . The area between and the -axis from to is ; the area between two curves is when . The volume of revolution about the -axis is .

Examples. . The curve for rotated about the -axis gives .

Differential equations and numerical methods

A first-order separable equation is solved by writing . Example: with : gives and since . Many real equations have no closed form, so numerical methods such as the trapezium rule with are used (Croft & Davison, 2019).

Common mistakes

Forgetting the constant of integration .

Forgetting the chain rule’s inner derivative.

Treating as when .

Forgetting to change the limits when using substitution in a definite integral.

Practice questions

Differentiate . []

Find . [1/2]

Find the stationary points of and classify them. [Maximum at , minimum at ]

Find the area between and . [1/6]