Calculus

Chapter 5 · Mathematics · 8 questions in the paper · 8 min read

Calculus

What the syllabus asks for 4
  • Limit and continuity of functions, indeterminate forms, L'Hospital's rule
  • Derivatives, rules of derivatives, geometrical and physical meanings, higher order derivatives, applications of derivative: tangent and normal, rate of change, maxima and minima
  • Integration, linear properties, rules of integration, standard integrals, definite integral, applications of definite integral: area under a curve and area between two curves
  • Differential equations: definition, order and degree, first order and first degree: variable separable, homogeneous, linear and exact equations, integrating factor

Calculus is, with algebra, the heaviest block of the paper (our estimate: about 8 questions). The official booklet's model questions include a hyperbolic-inverse derivative and a first-order linear differential equation, so the syllabus is read broadly: standard limits, derivatives of inverse and hyperbolic functions, tangent / rate / extremum problems, a definite integral or an area, and a first-order differential equation. Speed comes from knowing the standard results cold.

What the paper asks from this chapter
  • Standard limits and L'Hôpital: \frac{\sin x}{x}, \frac{1-\cos x}{x^2}, \frac{e^x-1}{x}, (1+x)^{1/x}.
  • Derivative table, including \sin^{-1},\tan^{-1}, \sinh^{-1},\cosh^{-1},\tanh^{-1},\mathrm{sech}^{-1}; parametric and implicit forms.
  • Tangent and normal, related rates, maxima and minima (second-derivative test).
  • Integrals: substitution, by parts, \int e^x(f+f')dx, Wallis, the property \int_0^af(x)dx=\int_0^af(a-x)dx.
  • Area under a curve and between two curves.
  • Differential equations: order, degree, separable, homogeneous, linear (integrating factor), exact.

5.1 Limit and continuity of functions, indeterminate forms, L'Hospital's rule

\lim_{x\to a}f(x)=L exists iff the left and right limits both equal L. f is continuous at a if \lim_{x\to a}f(x)=f(a). Polynomials, e^x, \sin x are continuous everywhere; rational functions except at zeros of the denominator.

Standard limit (x\to0) Value
\dfrac{\sin x}{x},\ \dfrac{\tan x}{x},\ \dfrac{\sin^{-1}x}{x},\ \dfrac{\tan^{-1}x}{x} 1
\dfrac{1-\cos x}{x^2} \frac{1}{2}
\dfrac{e^x-1}{x},\ \dfrac{\ln(1+x)}{x} 1
\dfrac{a^x-1}{x} \ln a
\dfrac{(1+x)^n-1}{x} n
(1+x)^{1/x}, and \big(1+\frac{1}{x}\big)^x as x\to\infty e
\dfrac{x^n-a^n}{x-a} (x\to a) na^{n-1}

Indeterminate forms: \frac{0}{0},\frac{\infty}{\infty},0\cdot\infty,\infty-\infty,1^\infty,0^0,\infty^0. L'Hôpital: for \frac{0}{0} or \frac{\infty}{\infty}, \lim\frac{f}{g}=\lim\frac{f'}{g'} (repeat if still indeterminate). Convert 0\cdot\infty to a quotient, \infty-\infty to a single fraction, and take logarithms for 1^\infty,0^0,\infty^0. Example: \lim_{x\to0}\frac{\tan x-\sin x}{x^3}=\lim\frac{\tan x(1-\cos x)}{x^3}=\frac{1}{2}. As x\to\infty the highest power of a rational function decides the limit.

Common mistake

L'Hôpital applies only to \frac{0}{0} or \frac{\infty}{\infty} forms. Using it on \frac{\sin x}{x} as x\to\infty is wrong (the limit is 0 by the squeeze theorem). For continuity of a piecewise function, set the left limit, the right limit and the value equal — and solve for the unknown constant.

5.2 Derivatives, rules, applications

Rules: sum, product (uv)'=u'v+uv', quotient \big(\frac{u}{v}\big)'=\frac{u'v-uv'}{v^2}, chain \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}; parametric \frac{dy}{dx}=\frac{dy/dt}{dx/dt}; implicit: differentiate both sides; logarithmic differentiation for x^x, u^v: \frac{d}{dx}x^x=x^x(1+\ln x).

f(x) f'(x) f(x) f'(x)
x^n nx^{n-1} \sin^{-1}x \frac{1}{\sqrt{1-x^2}}
e^x, a^x e^x, a^x\ln a \tan^{-1}x \frac{1}{1+x^2}
\ln x, \log_ax \frac{1}{x}, \frac{1}{x\ln a} \sec^{-1}x \frac{1}{|x|\sqrt{x^2-1}}
\sin x, \cos x \cos x, -\sin x \sinh x, \cosh x \cosh x, \sinh x
\tan x, \cot x \sec^2x, -\csc^2x \tanh x \mathrm{sech}^2x
\sec x, \csc x \sec x\tan x, -\csc x\cot x \sinh^{-1}x \frac{1}{\sqrt{x^2+1}}
\ln(\sec x+\tan x) \sec x \cosh^{-1}x (x>1) \frac{1}{\sqrt{x^2-1}}
\sqrt{x} \frac{1}{2\sqrt{x}} \tanh^{-1}x (|x|<1) \frac{1}{1-x^2}
\mathrm{sech}^{-1}x (0<x<1) \frac{-1}{x\sqrt{1-x^2}}

Hyperbolic: \sinh x=\frac{e^x-e^{-x}}{2}, \cosh x=\frac{e^x+e^{-x}}{2}, \cosh^2x-\sinh^2x=1; \sinh^{-1}x=\ln(x+\sqrt{x^2+1}), \cosh^{-1}x=\ln(x+\sqrt{x^2-1}), \tanh^{-1}x=\frac{1}{2}\ln\frac{1+x}{1-x}. The derivative of \mathrm{sech}^{-1}x=\cosh^{-1}(1/x) exists only inside its domain 0<x<1 — the reason the booklet's model answer is "x<1".

Higher derivatives and Leibniz: \frac{d^n}{dx^n}e^{ax}=a^ne^{ax}, \frac{d^n}{dx^n}\sin x=\sin(x+\frac{n\pi}{2}). Rolle (continuous on [a,b], differentiable inside, f(a)=f(b) gives f'(c)=0) and Mean value theorem f'(c)=\frac{f(b)-f(a)}{b-a}.

Applications

  • Tangent at (x_1,y_1): slope f'(x_1); normal slope -\frac{1}{f'(x_1)}. A horizontal tangent has f'=0.
  • Rate of change: differentiate the relation with respect to time. Sphere: V=\frac{4}{3}\pi r^3\Rightarrow\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}. Ladder: x^2+y^2=L^2\Rightarrow x\dot x+y\dot y=0.
  • Maxima and minima. Critical points: f'(x)=0. Second-derivative test: f''<0 local max, f''>0 local min. For a closed interval compare the critical values with the endpoint values. x^2-4x+7 has minimum 3 at x=2. Given perimeter or sum, a rectangle / product is maximal when the parts are equal.
  • Increasing where f'>0, decreasing where f'<0; concave up where f''>0; a point of inflection where f'' changes sign.

5.3 Integration, definite integral, area

Standard integrals (+C omitted): \int x^n=\frac{x^{n+1}}{n+1} (n\ne-1), \int\frac{1}{x}=\ln|x|, \int e^{ax}=\frac{e^{ax}}{a}, \int a^x=\frac{a^x}{\ln a}, \int\sin=-\cos, \int\cos=\sin, \int\sec^2=\tan, \int\csc^2=-\cot, \int\sec\tan=\sec, \int\tan x=\ln|\sec x|, \int\sec x=\ln|\sec x+\tan x|.

\int\frac{dx}{x^2+a^2}=\frac{1}{a}\tan^{-1}\frac{x}{a},\qquad \int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}.

\int\frac{dx}{x^2-a^2}=\frac{1}{2a}\ln\Big|\frac{x-a}{x+a}\Big|,\qquad \int\frac{dx}{\sqrt{x^2\pm a^2}}=\ln\big|x+\sqrt{x^2\pm a^2}\big|.

\int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}. \int\frac{f'(x)}{f(x)}dx=\ln|f|.

Methods. Substitution (\int\frac{x}{x^2+1}dx=\frac{1}{2}\ln(x^2+1)). By parts \int uv'=uv-\int u'v, with u chosen by ILATE (Inverse, Log, Algebraic, Trig, Exponential): \int xe^xdx=xe^x-e^x; \int\ln x\,dx=x\ln x-x. Special: \int e^x[f(x)+f'(x)]dx=e^xf(x). Partial fractions for rational functions.

Definite integral. \int_a^bf=F(b)-F(a). Properties: \int_a^bf(x)dx=\int_a^bf(a+b-x)dx; \int_{-a}^af=2\int_0^af if f even, 0 if odd; \int_0^{\pi/2}\frac{\sin x}{\sin x+\cos x}dx=\frac{\pi}{4} (also with \sin^n,\cos^n); Wallis: \int_0^{\pi/2}\sin^nx\,dx=\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots (times \frac{\pi}{2} if n even). \int_0^{\pi/2}\sin^2x\,dx=\frac{\pi}{4}.

Area. Under y=f(x) from a to b: \int_a^b|f|\,dx; between curves: \int_a^b(f-g)\,dx where f\ge g; with respect to y: \int x\,dy. Circle area \pi r^2, area of an ellipse \pi ab. Area between y^2=4ax and x^2=4ay: \frac{16a^2}{3}. Intersect the curves first to find the limits.

5.4 Differential equations

Order = highest derivative present; degree = its power after removing radicals and fractions of derivatives. \big(\frac{d^2y}{dx^2}\big)^3+\big(\frac{dy}{dx}\big)^4+y=0 has order 2, degree 3.

Type Form Method
separable \frac{dy}{dx}=f(x)g(y) \int\frac{dy}{g}=\int f\,dx
homogeneous \frac{dy}{dx}=F\big(\frac{y}{x}\big) put y=vx
linear \frac{dy}{dx}+Py=Q IF =e^{\int P\,dx}; y\cdot\text{IF}=\int Q\cdot\text{IF}\,dx
exact M\,dx+N\,dy=0, \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} \int M\,dx (y const) + terms of N free of x =C

If M\,dx+N\,dy=0 is not exact, try an integrating factor: e^{\int\frac{M_y-N_x}{N}dx} when that depends only on x. Examples: \frac{dy}{dx}+\frac{2}{x}y=x^2 has IF x^2; \frac{dy}{dx}+y=e^{-x} gives y=(x+C)e^{-x}. The number of arbitrary constants in the general solution equals the order.

Formula and fact sheet

Topic Result
\lim\frac{\sin x}{x}, \lim\frac{1-\cos x}{x^2} 1, \frac{1}{2}
\lim(1+x)^{1/x} e
\frac{d}{dx}\sinh^{-1}x \frac{1}{\sqrt{x^2+1}}
normal slope -1/f'
\int e^x(f+f') e^xf
\int\frac{dx}{a^2+x^2} \frac{1}{a}\tan^{-1}\frac{x}{a}
area between curves \int(f-g)\,dx
linear ODE IF e^{\int P\,dx}

Chapter summary

  • Standard limits plus L'Hôpital settle every limit; check the form before using L'Hôpital.
  • Chain rule, parametric and logarithmic differentiation cover all derivative types.
  • Critical points from f'=0; the sign of f'' decides max or min; compare endpoints on closed intervals.
  • Integrate by substitution, by parts (ILATE) or by the e^x(f+f') pattern; use symmetry for definite integrals.
  • Areas: find the intersections, then integrate top minus bottom.
  • Classify a first-order equation (separable, homogeneous, linear, exact) before solving.

Practice questions

Set A — Recall (14 questions)

  1. The value of \lim_{x\to0}\dfrac{\sin3x}{x} isa1b3c1/3d0
  2. The value of \lim_{x\to0}(1+2x)^{1/x} isaeb2ece^2d1
  3. \dfrac{d}{dx}\tan^{-1}x equalsa\dfrac{1}{1+x^2}b\dfrac{1}{\sqrt{1-x^2}}c-\dfrac{1}{1+x^2}d\sec^2x
  4. \displaystyle\int e^{2x}dx equalsa2e^{2x}+Cbe^{2x}+Cc\dfrac{e^x}{2}+Cd\dfrac{e^{2x}}{2}+C
  5. The value of \displaystyle\int_0^{\pi}\sin x\,dx isa2b0c1d\pi
  6. The order and degree of \big(\frac{d^2y}{dx^2}\big)^3+\big(\frac{dy}{dx}\big)^4+y=0 area3, 2b2, 3c2, 1d3, 3
  7. The general solution of \dfrac{dy}{dx}=y isay=Cxby=x+Ccy=Ce^xdy=C\ln x
  8. The integrating factor of \dfrac{dy}{dx}+\dfrac{2}{x}y=x^2 isa2xbe^{2x}c\ln x^2dx^2
  9. \dfrac{d}{dx}\big(e^x\sin x\big) equalsae^x\cos xbe^x(\sin x+\cos x)ce^x(\sin x-\cos x)d-e^x\sin x
  10. The slope of the tangent to y=x^3-3x at x=2 isa3b6c9d12
  11. The local maximum value of f(x)=3x-x^3 isa2b-2c0d4
  12. The value of \lim_{x\to0}\dfrac{1-\cos x}{x^2} isa0b1c2d1/2
  13. The area under y=x^2 from x=0 to x=3 isa3b9c27d6
  14. The value of \displaystyle\int_0^1xe^x\,dx isa0bec1de-1
Answers and explanations — Set A

1. (b) — \frac{\sin3x}{x}=3\cdot\frac{\sin3x}{3x}\to3.

2. (c) — (1+2x)^{1/x}=[(1+2x)^{1/2x}]^2\to e^2.

3. (a) — standard derivative.

4. (d) — \frac{e^{2x}}{2}; (a) differentiates instead.

5. (a) — [-\cos x]_0^\pi=1+1=2.

6. (b) — highest derivative is the second (order 2), raised to the power 3 (degree 3).

7. (c) — separate: \frac{dy}{y}=dx, \ln y=x+c.

8. (d) — e^{\int\frac{2}{x}dx}=e^{2\ln x}=x^2.

9. (b) — product rule.

10. (c) — 3x^2-3=12-3=9.

11. (a) — f'=3-3x^2=0 at x=\pm1; f''=-6x<0 at x=1, f(1)=2.

12. (d) — \frac{1-\cos x}{x^2}\to\frac{1}{2}.

13. (b) — \frac{x^3}{3}\Big|_0^3=9.

14. (c) — by parts: [xe^x-e^x]_0^1=(e-e)-(0-1)=1.

Set B — Understanding (10 questions)

  1. The value of \lim_{x\to0}\dfrac{\tan x-\sin x}{x^3} isa1/2b1c0d1/3
  2. The function f(x)=\dfrac{x^2-4}{x-2} (x\ne2) with f(2)=k is continuous at 2 when k=a0b2c4dno value
  3. If y=\ln(\sec x+\tan x), then \dfrac{dy}{dx} equalsa\tan xb\sec^2xc\sec x\tan xd\sec x
  4. If x=t^2 and y=t^3, then \dfrac{dy}{dx} at t=2 isa6b3c12d3/2
  5. The radius of a sphere grows at 2 cm/s. When r=5 cm the volume grows ata100\pib50\pic200\pi cm³/sd400\pi
  6. The minimum value of x^2-4x+7 isa7b4c2d3
  7. \displaystyle\int\frac{x}{x^2+1}dx equalsa\ln(x^2+1)+Cb\tfrac{1}{2}\ln(x^2+1)+Cc\tan^{-1}x+Cd2\ln(x^2+1)+C
  8. The value of \displaystyle\int_0^{\pi/2}\sin^2x\,dx isa\pi/4b\pi/2c1d\pi/8
  9. The area enclosed between y=x and y=x^2 isa1/2b1/3c1/6d1/12
  10. The solution of \dfrac{dy}{dx}=\dfrac{x+y}{x} isay=x(\ln x+C)by=x\ln x+Ccy=x+C\ln xdy=Cx^2
Answers and explanations — Set B

1. (a) — \tan x-\sin x=\tan x(1-\cos x)\approx x\cdot\frac{x^2}{2}.

2. (c) — \frac{x^2-4}{x-2}=x+2\to4; continuity needs k=4.

3. (d) — \frac{\sec x\tan x+\sec^2x}{\sec x+\tan x}=\sec x.

4. (b) — \frac{3t^2}{2t}=\frac{3t}{2}=3 at t=2.

5. (c) — \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}=4\pi\cdot25\cdot2=200\pi.

6. (d) — (x-2)^2+3.

7. (b) — substitute u=x^2+1.

8. (a) — \int\frac{1-\cos2x}{2}=\frac{1}{2}\big[\frac{\pi}{2}\big]=\frac{\pi}{4}.

9. (c) — \int_0^1(x-x^2)dx=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}.

10. (a) — homogeneous: y=vx gives xv'=1, v=\ln x+C.

Set C — Application (8 questions)

  1. The value of \lim_{x\to0}\dfrac{e^x-1-x}{x^2} isa0b1c2d1/2
  2. The maximum value of f(x)=xe^{-x} for x>0 isa1/ebec1d1/(2e)
  3. The normal to y=x^2 at (1,1) isa2x+y-3=0bx+2y-3=0cx-2y+1=0d2x-y-1=0
  4. The value of \displaystyle\int_0^{\pi/2}\dfrac{\sin x}{\sin x+\cos x}dx isa\pi/2b1c\pi/8d\pi/4
  5. The area enclosed by y^2=4x and y=x isa16/3b4c8/3d32/3
  6. A 5 m ladder slides with its foot moving away from the wall at 3 m/s. When the foot is 3 m from the wall, the top slides down at (m/s)a4/3b9/4c3/4d4
  7. The general solution of \dfrac{dy}{dx}+y=e^{-x} isay=(x+C)e^{-x}by=(x+C)e^{x}cy=e^{-x}+Cdy=xe^x+C
  8. The integrating factor of \dfrac{dy}{dx}+\dfrac{x}{1+x}y=1+x isae^xb(1+x)e^xc\dfrac{e^x}{1+x}d\dfrac{e^x}{x}
Answers and explanations — Set C

1. (d) — e^x-1-x=\frac{x^2}{2}+\frac{x^3}{6}+\cdots.

2. (a) — f'=e^{-x}(1-x)=0 at x=1; f(1)=1/e and f''(1)<0.

3. (b) — tangent slope 2, normal slope -\frac{1}{2}: y-1=-\frac{1}{2}(x-1).

4. (d) — I=\int_0^{\pi/2}\frac{\cos x}{\sin x+\cos x}dx by x\to\frac{\pi}{2}-x; 2I=\frac{\pi}{2}.

5. (c) — meet at x=0,4: \int_0^4(2\sqrt{x}-x)dx=\frac{32}{3}-8=\frac{8}{3}.

6. (b) — x^2+y^2=25: x\dot x+y\dot y=0 with x=3, y=4: 9+4\dot y=0, |\dot y|=\frac{9}{4}.

7. (a) — IF =e^x: (ye^x)'=1, so ye^x=x+C.

8. (c) — \int\frac{x}{1+x}dx=x-\ln(1+x), so IF =\frac{e^x}{1+x}.