Chapter 10. Introduction to Calculus

Limits. The limit of f(x) as x approaches a is the value f(x) gets close to as x gets close to a. A function is continuous at a if its limit equals f(a) [7][13].

Derivative. The derivative measures instantaneous rate of change and the slope of the tangent line:

f′(x) = lim (h → 0) [f(x + h) − f(x)] / h.

Basic rules

Constant: d/dx (c) = 0

Power rule: d/dx (xⁿ) = nxⁿ⁻¹

Sum: (f + g)′ = f′ + g′

Product: (fg)′ = f′g + fg′

Quotient: (f/g)′ = (f′g − fg′)/g²

Chain rule: d/dx f(g(x)) = f′(g(x)) · g′(x)

d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x

Applications. Setting f′(x) = 0 finds critical points, which help locate maxima and minima. Derivatives also give velocity (the derivative of position) and acceleration (the derivative of velocity).

Integral. The definite integral of f from a to b is the signed area under the curve. The Fundamental Theorem of Calculus links the two ideas: if F′ = f, then the integral of f from a to b equals F(b) − F(a) [7][13].

Basic antiderivatives: ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1); ∫ eˣ dx = eˣ + C; ∫ cos x dx = sin x + C; ∫ 1/x dx = ln|x| + C.

Review: (a) Differentiate x³ + 2x. (Answer: 3x² + 2.) (b) Evaluate the integral of 2x from 0 to 3. (Answer: 9.)