Limits and Differentiation

Calculus is 6–8 of the 50 questions, shared between this chapter and the next. The questions are rarely deep: a standard limit, a derivative by the product or quotient rule, a slope of a tangent, a maximum or minimum.

The Dean's Office model asks \lim_{x\to0}\frac{|\sin x|}{x} (it does not exist) and the derivative of x^2\cos x. Know the standard limits and the derivative table and these are quick.

12.1 Limits of algebraic and trigonometric functions

• Substitute first. If you get \frac{0}{0}, factor and cancel, or use a standard limit.
• \lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}; \lim_{x\to0}\frac{\sin x}{x}=\lim_{x\to0}\frac{\tan x}{x}=1; \lim_{x\to0}\frac{\sin kx}{x}=k; \lim_{x\to0}\frac{1-\cos x}{x^2}=\frac{1}{2}.
• \lim_{x\to0}\frac{e^x-1}{x}=1, \lim_{x\to0}\frac{\ln(1+x)}{x}=1, \lim_{x\to0}(1+x)^{1/x}=e.
• At infinity divide by the highest power: \lim_{x\to\infty}\frac{3x^2+2x}{x^2+5}=3.
• A limit exists only if the left and right limits agree: \frac{|\sin x|}{x} tends to -1 from the left and +1 from the right, so it does not exist.

12.2 Continuity

f is continuous at a if \lim_{x\to a}f(x)=f(a). To make f(x)=\frac{x^2-9}{x-3} continuous at 3, define f(3)=6 (the limit).

Product: (uv)'=u'v+uv'. Quotient: \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}. Chain: \frac{d}{dx}f(g(x))=f'(g(x))g'(x).

Example \frac{d}{dx}(x^2\cos x)=2x\cos x-x^2\sin x.

(a) Leibniz: the notation dy/dx

(b) Newton: fluxions

(c) Speedometer: rate of change of distance

12.5 Eight traps in limits and differentiation

Calculus questions in the entrance paper are mostly one-step rules, so the trap is almost always a missed rule or a value substituted too early. These eight traps cover the model questions on limits, derivatives and maxima.

1. Substitute first, and treat \frac00 as a signal. If direct substitution gives a number, that number is the limit. If it gives \frac00, factor and cancel, rationalise or use a standard limit: \frac{x^2-9}{x-3}=x+3 so the limit at 3 is 6 (Fig 12.1). A limit describes where f is heading, not what f equals at the point.
2. Know the standard limits and their conditions. \lim_{x\to0}\frac{\sin x}{x}=1 only for x in radians and x\to0, and \lim_{x\to0}\frac{\sin kx}{x}=k. Also \lim\frac{1-\cos x}{x^2}=\frac12, \lim\frac{e^x-1}{x}=1 and \lim_{x\to\infty}\left(1+\frac1x\right)^x=e. A common slip is to say \lim\frac{\sin x}{x}=1 as x\to\infty, when it is 0.
3. At infinity, divide by the highest power. For \frac{3x^2+2x}{x^2+5} the limit is 3. If the numerator's degree is lower the limit is 0, and if it is higher the limit is infinite. Do not cancel the leading terms without dividing the whole expression.
4. Check both sides and the value for continuity. A limit exists only if the left and right limits agree, so \frac{|x|}{x} has no limit at 0. A function is continuous at a when f(a) exists, \lim_{x\to a}f(x) exists and the two are equal. To repair \frac{x^2-9}{x-3} define f(3)=6.
5. Apply the right rule, and the chain rule every time. Product: (uv)'=u'v+uv', quotient: \frac{u'v-uv'}{v^2} with the order u'v-uv', and chain: differentiate the outside, then multiply by the derivative of the inside. So \frac{d}{dx}\sin2x=2\cos2x, \frac{d}{dx}a^x=a^x\ln a and \frac{d}{dx}\tan x=\sec^2x. The derivative of a constant is 0.
6. Tangent uses f'(a); normal uses its negative reciprocal. The tangent at x=1 on y=x^2 has slope 2 and equation y=2x-1 (Fig 12.2). The normal has slope -\frac12. Check that the tangent passes through the point of contact.
7. f'=0 finds candidates; f'' decides. f''<0 means a maximum and f''>0 means a minimum (Fig 12.3). If f''=0 the test fails: y=x^3 has f'(0)=0 but no turning point. On a closed interval also compare the end-point values.
8. For rates, link the variables by the chain rule, and keep units. \frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}, so a ripple of radius 5 cm growing at 2 cm/s has \frac{dA}{dt}=62.83 cm^2/s (Fig 12.4). Write the quantity to optimise in a single variable before differentiating. For two numbers with a fixed sum, the product is greatest when they are equal.

12.5.1 Worked checks

Q. Evaluate \lim_{x\to2}\frac{x^2-4}{x-2} and \lim_{x\to0}\frac{\sin3x}{x}.

Answer. Factor: \frac{(x-2)(x+2)}{x-2}=x+2\to4. For the second use \lim\frac{\sin kx}{x}=k, so the limit is 3. Both would give \frac00 if substituted straight in.

Q. Differentiate y=x^2\cos x and find the tangent to y=x^2 at x=1.

Answer. Product rule: y'=2x\cos x-x^2\sin x. For the tangent, f'(1)=2 and f(1)=1, so y-1=2(x-1), i.e. y=2x-1. This is the red line in Fig 12.2.

Q. Two positive numbers have sum 20. Find the greatest possible product.

Answer. Let the numbers be x and 20-x, so P=20x-x^2 and P'=20-2x=0 gives x=10. P''=-2<0, so this is a maximum, P=10\times10=100. Check: 9\times11=99<100.

12.4 Tangents, maxima, minima and rates

• Slope of the tangent at x=a is f'(a); the tangent is parallel to y=mx+c where f'(x)=m.
• Stationary points: f'(x)=0. Then f''>0 gives a minimum, f''<0 a maximum. f'>0 means increasing.
• Rates: \frac{dA}{dt}=\frac{dA}{dr}\cdot\frac{dr}{dt}. For a circle \frac{dA}{dr}=2\pi r; for a sphere \frac{dV}{dr}=4\pi r^2.
• Two numbers with sum S have maximum product \left(\frac{S}{2}\right)^2.

Practice questions

Set A — Recall (20 questions)

1. \lim_{x\to0}\frac{\sin x}{x} equalsa0binfinityc1d-1
2. \frac{d}{dx}(x^3) equalsa3x^2bx^2c3xd\frac{x^4}{4}
3. \frac{d}{dx}(\sin x) equalsa-\cos xb\sin xc-\sin xd\cos x
4. \frac{d}{dx}(\ln x) equalsaxb\frac{1}{x}c\ln xde^x
5. \frac{d}{dx}(e^x) equalsae^xbxe^{x-1}c1d\ln x
6. \lim_{x\to2}\frac{x^2-4}{x-2} equalsa0b2c4ddoes not exist
7. \frac{d}{dx}(\cos x) equalsa\sin xb\cos xc-\cos xd-\sin x
8. The derivative of a constant isa1b0cthe constant itselfdinfinity
9. \lim_{x\to\infty}\frac{1}{x} equalsainfinityb1c0d-1
10. \frac{d}{dx}(\tan x) equalsa\sec^2xb\cot xc-\csc^2xd\sec x\tan x
11. If f(x)=x^2, then f'(3) isa9b3c2d6
12. f has a local minimum at x=a if f'a=0 and f''(a) is (a) negativebpositiveczerodundefined
13. \lim_{x\to0}\frac{e^x-1}{x} equalsa1b0cedinfinity
14. \frac{d}{dx}(5x^2+3x) equalsa5x+3b10xc10x+3d10x^2+3
15. The slope of the tangent to y=x^2 at x=2 isa2b8c1d4
16. \frac{d}{dx}(uv) equalsau'v'bu'v+uv'cu'v-uv'd\frac{u'v-uv'}{v^2}
17. \lim_{x\to0}(1+x)^{1/x} equalsa1b0cedinfinity
18. \frac{d}{dx}\left(\frac{1}{x}\right) equalsa-\frac{1}{x^2}b\frac{1}{x^2}c\ln xd-\frac{1}{x}
19. \lim_{x\to0}\frac{\tan x}{x} equalsa0binfinityc-1d1
20. \frac{d}{dx}(\sqrt{x}) equalsa\frac{1}{\sqrt{x}}b\frac{1}{2\sqrt{x}}c2\sqrt{x}d\frac{\sqrt{x}}{2}

Set B — Understanding (12 questions)

1. \lim_{x\to0}\frac{\sin3x}{x} equalsa1b3c\frac{1}{3}d0
2. \frac{d}{dx}(x^2\cos x) equalsa2x\cos x+x^2\sin xb-2x\sin xc2x\cos xd2x\cos x-x^2\sin x
3. \lim_{x\to\infty}\frac{3x^2+2x}{x^2+5} equalsa2b0c3dinfinity
4. \lim_{x\to0}\frac{1-\cos x}{x^2} equalsa\frac{1}{2}b1c0d2
5. \lim_{x\to1}\frac{x^3-1}{x-1} equalsa0b1c2d3
6. \frac{d}{dx}\left(\frac{x}{x+1}\right) equalsa\frac{1}{x+1}b\frac{1}{(x+1)^2}c\frac{x}{(x+1)^2}d-\frac{1}{(x+1)^2}
7. The minimum value of x^2-4x+7 isa3b7c2d4
8. f(x)=x^3-3x has a local maximum at x=a1b0c-1d3
9. The tangent to y=x^3 at (1,1) meets the y-axis ata(0,2)b(0,-2)c(0,3)d(0,-3)
10. The rate of change of the area of a circle with respect to its radius at r=5 isa25\pib5\pic\pid10\pi
11. \lim_{x\to0}\frac{|\sin x|}{x}adoes not existbequals 1cequals -1dequals 0
12. f(x)=\frac{x^2-9}{x-3} for x\ne3 is continuous at 3 if f(3) equalsa0b3c6d9

Set C — Application (8 questions)

1. \lim_{x\to0}\frac{\sin5x}{\sin2x} equalsa\frac{2}{5}b1c3d\frac{5}{2}
2. Two positive numbers have sum 20. Their greatest possible product isa100b96c99d120
3. A spherical balloon's radius grows at 2 cm/s. At r=3 cm the volume grows ata36\pi cm³/sb24\pi cm³/sc72\pi cm³/sd18\pi cm³/s
4. The local maximum value of x^3-6x^2+9x isa0b4c9d3
5. If y=\sin x\cos x, then \frac{dy}{dx} at x=0 isa0b\frac{1}{2}c-1d1
6. \frac{d}{dx}(xe^x) at x=0 isa0bec1d2
7. The tangent to y=x^2 is parallel to y=4x+1 at the pointa(2,4)b(4,16)c(1,1)d(-2,4)
8. f(x)=2x^3-9x^2+12x is decreasing onax<1b1<x<2cx>2dall real numbers