Chapter 6 — Applications Of Derivatives

What it means
The rate of change of a quantity describes how that quantity changes with respect to another (usually time or space). In calculus this is measured by the derivative. If y = f(x), then dy/dx gives the instantaneous rate of change of y with respect to x. Units of dy/dx are (units of y)/(units of x).

Average vs Instantaneous rate

• Average rate of change of y between x = a and x = b: (f(b) - f(a)) / (b - a).
• Instantaneous rate of change at x: limit of average rates as b → a, i.e. f'(x) = lim(h→0) [f(x+h)-f(x)]/h.

Interpretation of sign and value

• f'(x) > 0: y is increasing at x. f'(x) < 0: y is decreasing.
• f'(x) = 0: horizontal tangent (possible max/min or inflection).
• Magnitude |f'(x)| gives steepness; large value means rapid change.

Related rates
Many problems involve two or more quantities that depend on a common variable (often time t). If z = g(x,y,...) and x, y depend on t, differentiate each relation with respect to t using the chain rule: dz/dt = (∂g/∂x)(dx/dt) + (∂g/∂y)(dy/dt) + ... . Implicit differentiation is used when variables are not solved explicitly.

Key methods used

• Direct differentiation: apply power, product, quotient and chain rules to get dy/dx or d/dt.
• Implicit differentiation: differentiate both sides of an equation containing x and y, then solve for dy/dx.
• Chain rule for rates: if y = f(u) and u = u(t), then dy/dt = f'(u)·du/dt.
• Higher derivatives: second derivative d2y/dx2 gives rate of change of the rate (e.g., acceleration = d2s/dt2).

Physical meaning & units
Always attach units: if s(t) is distance in metres and t in seconds, ds/dt (m/s) is velocity; d2s/dt2 (m/s2) is acceleration. Sign tells direction (positive/negative).

Common pitfalls

• Differentiate with respect to the correct variable (dx/dt vs dy/dx).
• When using geometry relations (e.g., volume vs radius), remember to differentiate the relation linking the quantities before substituting rates.

Summary
Rate-of-change problems reduce to: identify dependent quantities, write an equation relating them, differentiate with respect to the common variable (often t) using chain/implicit rules, then plug known values to find the unknown rate.

Definition: Let f be a real-valued function defined on an interval I. f is called increasing on I if for every x1, x2 in I with x1 < x2 we have f(x1) < f(x2). It is called decreasing on I if for every x1 < x2 we have f(x1) > f(x2). (If < or > are replaced by ≤ or ≥, we say non-decreasing or non-increasing.)

Derivative test (differentiable functions):

• If f is differentiable on an interval and f'(x) > 0 for every x in that interval, then f is increasing on that interval.
• If f is differentiable on an interval and f'(x) < 0 for every x in that interval, then f is decreasing on that interval.

Notes and subtleties:

• If f is increasing, then wherever f'(x) exists we must have f'(x) ≥ 0, but the converse need not be strict: f'(x) can be 0 at isolated points while f remains increasing (example: f(x)=x^3 has f'(0)=0 but is increasing everywhere).
• Critical points are points in the domain where f'(x)=0 or f'(x) does not exist. These points partition the domain into intervals on which the derivative sign is constant; this sign determines monotonicity on each interval.
• First derivative test for local extrema: If f' changes sign from + to − at c, f has a local maximum at c. If f' changes sign from − to + at c, f has a local minimum at c. If no sign change occurs, there is no local extremum at c (it might be a saddle/inflection point).

Procedure to determine increasing/decreasing intervals:

1. Find domain of f.
2. Compute f'(x) and solve f'(x)=0 and find where f' is undefined to get critical points.
3. Make a sign chart (number line) using the critical points and test the sign of f' on each subinterval.
4. Conclude: f is increasing where f'>0 and decreasing where f'<0. Mark local extrema using sign change.

Common pitfalls: Don't conclude f is decreasing at a point where f' is negative only at isolated points; monotonicity is about intervals. Also check domain—log, root, rational functions may have restricted domains.

Overview: A tangent to a curve at a point is the straight line that best approximates the curve near that point. The slope of the tangent equals the derivative of the function at that point. A normal at the same point is the line perpendicular to the tangent.

Tangent (for y = f(x)): If the curve y = f(x) is differentiable at x = x1, the slope of the tangent at P(x1, y1), where y1 = f(x1), is m_t = f'(x1). The equation of the tangent line is

y - y1 = f'(x1) (x - x1)

Normal: The normal is perpendicular to the tangent. If f'(x1) is finite and nonzero, the slope of the normal is m_n = -1 / f'(x1). The equation of the normal is

y - y1 = -1/f'(x1) (x - x1)

Implicit and parametric forms:

• Implicit curve F(x,y)=0: use implicit differentiation. dy/dx = -F_x / F_y (provided F_y ≠ 0).
• Parametric curve x = x(t), y = y(t): slope dy/dx = (dy/dt) / (dx/dt) and then use tangent/normal equations in parametric form.

Geometric/analytic facts:

• Angle between two lines with slopes m1 and m2: tan(theta) = (m2 - m1)/(1 + m1 m2) when denominator ≠ 0.
• For a level curve g(x,y)=c, the gradient vector ∇g = (g_x, g_y) is normal to the curve at each point.
• Linear approximation (tangent as best linear estimate): f(x) ≈ f(x1) + f'(x1)(x - x1) for x near x1.

Applications: instant velocity (position-time graph tangent), small-change approximations in physics and economics, normal direction in optics (law of reflection uses normal to surface), normals in computer graphics for shading, and normals for machining/surface design.

Idea: If a function f is differentiable at a point a, then for small changes x = a + dx the function value f(x) can be approximated by the value of the tangent line to f at a. This first‑order approximation is called linearization.

Linearization formula: Let L(x) denote the linearization of f at a. Then

L(x) = f(a) + f'(a)(x - a).

Writing dx = x - a and dy to denote the differential change in y, we get the differential relation

dy = f'(a) dx, so f(a + dx) ≈ f(a) + dy = f(a) + f'(a) dx.

When it holds: This approximation is good when dx is small and f is differentiable at a. The linearization is the first two terms of the Taylor expansion of f around a.

Error and bound: By Taylor's theorem with remainder (Lagrange form), for some ξ between a and a + dx,

f(a + dx) = f(a) + f'(a) dx + (1/2) f''(ξ) (dx)^2.

Hence the error (actual − linear) satisfies

|Error| = |f(a + dx) − L(a + dx)| = |(1/2) f''(ξ)| |dx|^2.

If M is a bound for |f''(t)| on the interval between a and a+dx, then

|Error| ≤ (M/2) (dx)^2. This shows error is of order (dx)^2, so for very small dx the linear approximation is accurate.

Relative change: For nonzero f(a), the relative change approximately satisfies

Δf / f(a) ≈ f'(a)/f(a) · dx.

Use in practice: Linearization is used to estimate values that are difficult to compute exactly, to estimate measurement-induced changes (propagation of errors), and to simplify nonlinear relationships in engineering, physics and economics for small perturbations.

What are maxima and minima? For a real-valued function f(x), a point x = c is a local maximum if f(c) ≥ f(x) for x near c, and a local minimum if f(c) ≤ f(x) for x near c. A global (absolute) maximum or minimum holds for all x in the domain. Points where f'(x)=0 or f' does not exist are called critical points.

Steps to find extrema (single-variable):

• Determine the domain of f. If the interval is closed and bounded, endpoints must be checked for absolute extrema.
• Compute f'(x) and solve f'(x)=0 and note where f' is undefined — these are critical points.
• Use a test to classify each critical point:
• Second derivative test: If f''(c)>0 → local minimum; if f''(c)<0 → local maximum; if f''(c)=0 → test inconclusive (use higher derivative test or sign chart).
• First derivative (sign) test: Check sign of f' to left and right of c: +→- gives max, -→+ gives min, no sign change gives no extremum.
• For constrained optimization (e.g. with a condition g(x,y)=0), use substitution or Lagrange multipliers: ∇f = λ∇g along with the constraint.
• Compare values of f at all candidate points (critical points and boundaries) to determine global maxima/minima.

Important remarks: f'(c)=0 is only a necessary condition for interior extrema, not sufficient. Inflection points (where concavity changes) may have f'(c)=0 but not extrema (example: f(x)=x^3 at 0).

Higher-derivative test (brief): If f'(c)=f''(c)=...=f^{(n-1)}(c)=0 and f^{(n)}(c)≠0, then if n is even: f has a local minimum at c if f^{(n)}(c)>0 and a local maximum if f^{(n)}(c)<0; if n is odd, no extremum (inflection).

Overview

Curve sketching using derivatives is the process of using first and second derivatives of a function f(x) to determine its important qualitative features — domain, intercepts, symmetry, asymptotes, intervals of increase/decrease, local maxima/minima, concavity, and points of inflection — and then drawing an accurate graph.

Step-by-step procedure

1. Domain and intercepts: Find the domain of f, x- and y-intercepts.
2. Symmetry: Check f(−x)=f(x) (even) or f(−x)=−f(x) (odd) to simplify sketching.
3. Asymptotes and end behaviour: Vertical asymptotes where f blows up (denominator = 0 for rational functions). Horizontal asymptote if lim_{x→±∞} f(x)=L. Oblique asymptote y=mx+c when f(x)≈mx+c as |x|→∞; compute m=lim_{x→∞} f(x)/x and c=lim_{x→∞} (f(x)-mx).
4. First derivative f'(x): Solve f'(x)=0 or undefined to find critical points. Use sign chart of f' to determine intervals of increase (f'>0) and decrease (f'<0) and to classify stationary points via the first derivative test.
5. Second derivative f''(x): Use f'' to determine concavity: f''>0 means concave up, f''<0 concave down. If f''(c)>0 at a stationary point c, it is a local minimum; if f''(c)<0, a local maximum (second derivative test).
6. Points of inflection: Solve f''(x)=0 or undefined and check for change of sign in f'' across the point.
7. Put together: Mark intercepts, critical points, inflection points, asymptotes and sketch with correct end behaviour and concavity. Use a derivative sign chart to support the sketch.

Useful remarks

• Stationary point with f'(c)=0 but no sign change in f' is a saddle point or point of inflection (example: x^3 at 0).
• Always check one-sided limits near vertical asymptotes to know whether the curve goes to +∞ or −∞.
• For rational functions, polynomial long division gives oblique asymptotes when degree(numerator) ≥ degree(denominator).

What is optimization? Optimization is the process of finding the largest or smallest value (maximum or minimum) of a function subject to given conditions. In Calculus, we use derivatives to locate and classify these extreme values.

Basic idea / steps to solve optimization problems (single variable)

• 1. Model the problem: express the quantity to be optimized as a function f(x) of one variable x and determine the domain.
• 2. Find critical points: compute f'(x) and solve f'(x)=0 and include points where f' is undefined but f is defined.
• 3. Test each candidate: use the second derivative test (if f''(c)>0 then local min, if f''(c)<0 then local max) or use the first derivative sign change. Also evaluate endpoints of a closed interval.
• 4. Interpret the result in the context of the problem (units, feasibility).

Two-variable optimization (unconstrained)

• Find stationary points by solving f_x(x,y)=0 and f_y(x,y)=0.
• Use the second derivative (Hessian) test: D = f_xx * f_yy - (f_xy)^2. If D>0 and f_xx>0, local minimum; if D>0 and f_xx<0, local maximum; if D<0, a saddle point; if D=0, test is inconclusive.

Constrained optimization (one constraint)

• Substitution: if constraint allows, express variables in terms of one variable and reduce to single-variable problem.
• Lagrange multipliers: solve ∇f = λ∇g together with g(x,y)=constant (useful when substitution is inconvenient).

Important practical notes

• Always check domain and endpoints (global maxima/minima on closed intervals may occur at endpoints).
• If second derivative test fails (f''=0), use higher derivatives or inspect sign change of f'.
• For physical problems include constraints such as non-negativity and integer requirements if applicable.

Overview: Rolle's Theorem and the Mean Value Theorem (MVT) are fundamental results in differential calculus that relate the behaviour of a function on an interval to the existence of points where its derivative takes specific values. They are widely used in proving inequalities, analysing functions and solving real-world problems involving rates.

Rolle's Theorem (statement): Let f be a function such that

• f is continuous on the closed interval [a, b],
• f is differentiable on the open interval (a, b),
• f(a) = f(b).

Then there exists at least one c in (a, b) such that f'(c) = 0 (i.e., a horizontal tangent exists).

Mean Value Theorem (MVT) (statement): Let f be a function such that

• f is continuous on [a, b],
• f is differentiable on (a, b).

Then there exists at least one c in (a, b) such that

f'(c) = (f(b) - f(a)) / (b - a).

Interpretation: some instantaneous rate of change (derivative) equals the average rate of change over [a, b].

Sketch of proof (MVT using Rolle):

1. Consider the secant line joining (a, f(a)) and (b, f(b)). Define g(x) = f(x) - L(x), where L(x) is the equation of this secant line.
2. Then g(a) = g(b). Also g is continuous on [a,b] and differentiable on (a,b).
3. By Rolle's theorem there exists c in (a,b) with g'(c) = 0. But g'(x) = f'(x) - L'(x), and L'(x) = (f(b)-f(a))/(b-a). Rearranging gives the MVT result.

Why the hypotheses matter:

• If continuity fails on [a,b], the function could jump and no guarantee of point c holds.
• If differentiability fails inside (a,b), like a sharp corner, the derivative may not exist at the required point (e.g., f(x)=|x| on [-1,1]; f(-1)=f(1) but no point with f'(c)=0).

Useful consequences and applications:

• Prove that if f'(x)=0 for all x in an interval, then f is constant on that interval.
• Estimate function growth and bound errors (Taylor remainder estimation uses MVT ideas).
• Relate average and instantaneous velocities in physics and engineering.