Difficulty: Intermediate to Advanced | Prerequisites: All derivative rules including chain rule and implicit differentiation (Ch. 2.1–2.6 notes).
This set covers Stewart sections 2.7 through 3.2. Here the derivative stops being an abstract calculation and becomes a tool for solving real problems: how fast is the water level rising in a conical tank? What is a good linear estimate of sqrt(1.1) without a calculator? Where does a function reach its highest or lowest value? The Mean Value Theorem (MVT) then provides the theoretical backbone connecting a function's derivative to its overall behaviour.
Related rates use the chain rule to link the rates of change of two or more variables that are connected by a geometric or physical equation. Linear approximation replaces a curve with its tangent line near a known point to estimate nearby values. Critical points (where f' = 0 or is undefined) locate potential maxima and minima, and the Mean Value Theorem guarantees that somewhere on an interval the instantaneous rate of change equals the average rate of change.
Related rates
Problems where two or more quantities change with time and are connected by an equation. You differentiate the equation with respect to time t (using the chain rule) to relate their rates. Think of it as: if the radius of a balloon is growing, how fast is the volume growing?
Linear approximation (tangent line approximation)
The estimate f(x) approximately equals f(a) + f'(a)(x - a) for x near a. In simple terms, you replace the curve with its tangent line to get a quick numerical estimate.
Differential
The quantity dy = f'(x) dx, representing the approximate change in y produced by a small change dx in x. Think of it as the tangent-line version of the actual change Delta y.
Absolute (global) maximum/minimum
The largest or smallest value of f(x) on its entire domain (or a specified interval). Also called the global extremum.
Local (relative) maximum/minimum
A value f(c) that is larger (or smaller) than f(x) for all x in some small neighbourhood of c, but not necessarily the largest on the whole domain.
Critical point (critical number)
A value x = c in the domain of f where f'(c) = 0 or f'(c) does not exist. Every local extremum occurs at a critical point (but not every critical point is an extremum).
Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum somewhere on [a, b].
Mean Value Theorem (MVT)
If f is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a). In simple terms, somewhere on the interval the slope of the tangent equals the slope of the secant.
Setup procedure:
Draw a picture and label all changing quantities with variables
Write an equation relating the variables (geometry, physics, etc.)
Differentiate both sides with respect to time t, applying the chain rule
Plug in the known values and rates to solve for the unknown rate
Example (ice block): a block with length l, width w, height h melts over time. V = lwh, so dV/dt = l'wh + lw'h + lwh'. With l = 10, w = 5, h = 20, l' = w' = -1, h' = -2: dV/dt = (-1)(5)(20) + (10)(-1)(20) + (10)(5)(-2) = -400 cm^3/hr
Common geometry formulas used: circle area A = pir^2, sphere volume V = (4/3)pir^3, cone volume V = (1/3)pir^2h, Pythagorean theorem, similar triangles
Always identify which rates are given and which rate you need to find before differentiating
Linear approximation: f(x) is approximately f(a) + f'(a)(x - a) for x near a
Example: estimate sqrt(1.1). Take f(x) = sqrt(x), a = 1. f(1) = 1, f'(x) = 1/(2*sqrt(x)), f'(1) = 1/2. So sqrt(1.1) is approximately 1 + (1/2)(0.1) = 1.05. Actual value: 1.04881...
Error analysis: the approximation error is roughly (1/2)|f''(c)|(x - a)^2 for some c between a and x. The closer x is to a, the better the approximation
Differential notation: dy = f'(x) dx approximates Delta y = f(x + dx) - f(x)
Extreme Value Theorem: a continuous function on a closed interval [a, b] must attain an absolute max and an absolute min
Fermat's Theorem: if f has a local extremum at c and f'(c) exists, then f'(c) = 0
Critical points: where f'(c) = 0 (horizontal tangent) or f'(c) does not exist (corner, cusp, vertical tangent)
Closed interval method for finding absolute extrema on [a, b]:
Find all critical points of f in (a, b)
Evaluate f at each critical point and at both endpoints a and b
The largest value is the absolute max; the smallest is the absolute min
Not every critical point is an extremum. f(x) = x^3 has f'(0) = 0 but x = 0 is an inflection point, not a max or min
Statement: if f is continuous on [a, b] and differentiable on (a, b), then there exists c in (a, b) with f'(c) = [f(b) - f(a)] / (b - a)
Geometric meaning: the tangent at some interior point is parallel to the secant line connecting (a, f(a)) and (b, f(b))
Rolle's Theorem is the special case where f(a) = f(b), giving f'(c) = 0 for some c in (a, b)
Consequences:
If f'(x) = 0 for all x in an interval, then f is constant on that interval
If f'(x) > 0 for all x in an interval, then f is increasing on that interval
If f'(x) = g'(x) for all x, then f(x) = g(x) + C for some constant C
Students often forget to differentiate the constraint equation with respect to time in related rates, treating variables as constants instead of functions of t.
Students assume every critical point is a max or min. f(x) = x^3 has a critical point at x = 0 that is neither.
Students sometimes check only critical points and forget the endpoints when finding absolute extrema on a closed interval. The endpoints are candidates too.
Students confuse the MVT conclusion (there exists some c) with knowing the exact value of c. The theorem guarantees existence, not a formula for c.
⚠️ Related rates: the four-step method (draw, write equation, differentiate, plug in) is tested on virtually every Calc I exam. Practise cones, expanding circles, ladders, and shadow problems.
⚠️ Linear approximation: "estimate without a calculator" questions. Know the formula and be ready to choose a good base point a.
⚠️ Closed interval method: finding absolute max/min on [a, b] is a standard problem. Do not skip the endpoints.
⚠️ MVT: typically tested as "show there exists a c" or "explain why the speedometer must have read X at some point."
In related rates, what rule do you apply when differentiating with respect to t? (The chain rule.)
True or false: the linear approximation of f(x) near a is f(a) + f'(a)(x - a). (True.)
Where can absolute extrema occur on a closed interval? (At critical points or at endpoints.)
If f'(x) > 0 on an interval, what does the MVT tell you about f? (f is increasing on that interval.)
True or false: Rolle's Theorem is a special case of the MVT where f(a) = f(b). (True.)
Q: A circular oil spill has radius growing at 2 m/min. How fast is the area increasing when r = 10 m?
A: A = pir^2, so dA/dt = 2pir(dr/dt) = 2pi(10)(2) = 40pi m^2/min.
Q: Use linear approximation to estimate (8.1)^(1/3).
A: Let f(x) = x^(1/3), a = 8. f(8) = 2, f'(x) = (1/3)x^(-2/3), f'(8) = 1/12. So (8.1)^(1/3) is approximately 2 + (1/12)(0.1) = 2.00833. Actual: 2.00831...
Q: Find the absolute maximum and minimum of f(x) = x^3 - 3x + 1 on [-2, 2].
A: f'(x) = 3x^2 - 3 = 3(x-1)(x+1) = 0 at x = 1 and x = -1. Evaluate: f(-2) = -1, f(-1) = 3, f(1) = -1, f(2) = 3. Absolute max = 3 at x = -1 and x = 2. Absolute min = -1 at x = -2 and x = 1.
Q: A car travels from city A to city B (100 km apart) in exactly 1 hour. Use the MVT to show the car's speed was exactly 100 km/h at some moment.
A: Let s(t) be position. s is continuous on [0, 1] and differentiable on (0, 1). By the MVT, there exists c in (0, 1) with s'(c) = [s(1) - s(0)] / (1 - 0) = 100/1 = 100 km/h.
Related rates combine implicit differentiation (Ch. 2.6) with the chain rule (Ch. 2.5). The closed-interval method for extrema leads directly into curve sketching and optimisation (Ch. 3.3 through 3.7). The MVT is the theoretical foundation for the Fundamental Theorem of Calculus (Ch. 4.3), connecting derivatives and integrals.
related rates, linear approximation, tangent line approximation, differential, absolute maximum, absolute minimum, local extremum, critical point, critical number, Fermat's theorem, Extreme Value Theorem, closed interval method, Mean Value Theorem, MVT, Rolle's Theorem, increasing function, decreasing function, MATH 132, Stewart Chapters 2.7 through 3.2, Calculus I