Applications of Differential Calculus — Study Notes (Class 12)
Applications of Differential Calculus — Study Notes
This chapter puts the first and second derivatives to work: reading slopes and rates from a function, locating tangents and normals, testing where a curve rises or falls, finding its high and low points, describing how it bends, and finally pulling all of this together to sketch curves and solve optimisation problems.
1. Derivative as slope and as rate
The derivative \( f'(x) \) has two faces. Geometrically it is the slope of the tangent at \( (x,f(x)) \); if \( \theta \) is the angle the tangent makes with the positive \( x \)-axis, then \( f'(x)=\tan\theta \). Physically it is an instantaneous rate of change. For motion on a line with position \( s(t) \), velocity is \( v=\frac{ds}{dt} \) and acceleration is \( a=\frac{dv}{dt}=\frac{d^{2}s}{dt^{2}} \). A particle is momentarily at rest when \( v=0 \), and it changes direction only where \( v \) changes sign.
2. Related rates
When several quantities vary with time and are linked by an equation, differentiate the whole equation with respect to \( t \) and substitute the instant of interest after differentiating. Typical links are the sphere \( V=\frac{4}{3}\pi r^{3} \), \( S=4\pi r^{2} \); the cone; and right-triangle relations \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \) for angle-of-elevation problems.
3. Tangent and normal
At \( (x_1,y_1) \) with slope \( m=f'(x_1) \), the tangent is \( y-y_1=m(x-x_1) \) and the normal (perpendicular to it) is \( y-y_1=-\frac{1}{m}(x-x_1) \). A horizontal tangent occurs where \( m=0 \); a vertical tangent occurs where \( m \) is undefined (in implicit problems, where the denominator of \( \frac{dy}{dx} \) vanishes).
4. Angle between two curves
The angle of intersection is the angle between the tangents at the common point: \[ \tan\varphi=\left|\frac{m_1-m_2}{1+m_1 m_2}\right|. \] The curves are orthogonal (cut at right angles) when \( m_1 m_2=-1 \), and touch when \( m_1=m_2 \).
5. Mean Value Theorems
Rolle's theorem: if \( f \) is continuous on \( [a,b] \), differentiable on \( (a,b) \), and \( f(a)=f(b) \), then some \( c\in(a,b) \) has \( f'(c)=0 \). Lagrange's MVT: dropping the condition \( f(a)=f(b) \), some \( c\in(a,b) \) has \[ f'(c)=\frac{f(b)-f(a)}{b-a}, \] i.e. the instantaneous rate equals the average rate. A consequence: if \( f'(x)=0 \) throughout an interval then \( f \) is constant there.
6. Series expansions
Taylor's series about \( x=a \): \[ f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}. \] Taking \( a=0 \) gives the Maclaurin series \( f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n} \).
7. Indeterminate forms and L'Hopital's rule
For \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), \[ \lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} \] provided the right side exists. Forms \( 0\cdot\infty \) and \( \infty-\infty \) are first rewritten as a single fraction; the power forms \( 0^{0},\,1^{\infty},\,\infty^{0} \) are handled by taking logarithms, finding the limit \( L \) of \( \ln y \), and exponentiating to get \( e^{L} \).
8. First derivative — monotonicity and local extrema
On an interval where \( f'(x)>0 \) the function is (strictly) increasing; where \( f'(x)<0 \) it is decreasing. A critical point is where \( f'(x)=0 \) or \( f' \) does not exist. By the First Derivative Test, at a critical point \( c \): a sign change \( -\to+ \) marks a local minimum, \( +\to- \) marks a local maximum, and no sign change means neither.
9. Absolute extrema on a closed interval
A continuous function on \( [a,b] \) attains both an absolute maximum and minimum. To find them: list the critical numbers inside \( (a,b) \), evaluate \( f \) at those numbers and at the endpoints \( a,b \), and pick the largest and smallest values.
10. Second derivative — concavity, inflection, extrema
If \( f''(x)>0 \) the graph is concave up; if \( f''(x)<0 \) it is concave down. A point of inflection is where concavity changes — \( f''=0 \) (or is undefined) and the sign of \( f'' \) actually switches there. By the Second Derivative Test, at a stationary point \( c \) (\( f'(c)=0 \)): \( f''(c)<0 \Rightarrow \) local maximum, \( f''(c)>0 \Rightarrow \) local minimum, and \( f''(c)=0 \) is inconclusive (fall back to the first derivative test).
11. Optimisation
Translate the problem into one variable using the constraint, write the quantity to be optimised, fix the valid domain, and apply an extremum test. Geometric distance problems are simplest if you optimise the squared distance.
12. Symmetry, asymptotes and curve sketching
A curve is symmetric about the \( y \)-axis if replacing \( x \) by \( -x \) leaves it unchanged, about the \( x \)-axis if replacing \( y \) by \( -y \) does, and about the origin if replacing both does. A line \( x=a \) is a vertical asymptote if \( y\to\pm\infty \) as \( x\to a \); \( y=b \) is a horizontal asymptote if \( y\to b \) as \( x\to\pm\infty \). A full sketch combines domain, intercepts, symmetry, asymptotes, monotonicity, extrema and concavity.
Common exam traps
- Tangent vs normal slope. The normal slope is \( -\frac{1}{m} \), not \( m \). Reversing them is the commonest slip (see the \( 2\cos 4x \) problem).
- Vertical tangents. Set the denominator of \( \frac{dy}{dx} \) to zero, not the numerator. For \( y^{2}-xy+9=0 \) this gives \( y=\pm 3 \), not \( \pm\sqrt{3} \).
- L'Hopital only on indeterminate forms. Re-applying the rule once a limit is no longer \( \frac00 \) or \( \frac{\infty}{\infty} \) gives wrong answers.
- Inflection needs a sign change. \( f''=0 \) alone is not enough; \( y=x^{4} \) has \( y''(0)=0 \) but no inflection there.
- Stationary vs critical. Every stationary point is critical, but a critical point where \( f' \) is undefined (e.g. \( x^{2/3} \) at \( 0 \)) is not stationary.
- Closed-interval extrema. Always test the endpoints, not just the interior critical numbers.
- Substitute after differentiating in related-rates and MVT problems — never before.