Lesson 12 of 12 · 11 min

Chapter review

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13 facts

1Average velocity = distance covered / time taken; instantaneous velocity is its limit as the interval shrinks.
2For s = 4.9t², the velocity at t = 2 s lies between 19.551 and 19.649 m/s.
3lim_{x→a} f(x) exists only if the left and right hand limits exist and are equal.
4A limit does not depend on f(a); f need not even be defined at a.
5Limits of sums, differences, products and quotients (non-zero denominator) are the sums, differences, products and quotients of limits.
6Polynomials: lim_{x→a} f(x) = f(a).
70/0 form: cancel the common factor (x − a) and evaluate again.
8lim_{x→a} (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹.
9lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1 − cos x)/x = 0, with x in radians.
10f′(a) = lim_{h→0} [f(a + h) − f(a)]/h: the slope of the tangent at (a, f(a)).
11(uv)′ = u′v + uv′ and (u/v)′ = (u′v − uv′)/v².
12d/dx xⁿ = nxⁿ⁻¹; the derivative of a constant is 0.
13d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec² x.

Common traps

Taking f(a) as the limit of a piecewise function.

Find the left and right hand limits from the pieces used near a. f(a) plays no part; in the textbook's example the limit is 3 while f(1) = 0.

Declaring a 0/0 limit undefined.

0/0 is a signal to simplify, not an answer. Cancel the vanishing factor, then substitute.

Cancelling a factor in the non-zero-over-zero case and getting a number.

If the numerator is non-zero and the denominator is zero at a, the limit does not exist.

Using degrees in sin x / x.

The limit is 1 only with x in radians; the area argument uses the radian measure of the sector.

Writing the quotient rule as (uv′ − u′v)/v².

The derivative of the numerator comes first: (u′v − uv′)/v². Test with u = 1, v = x, which must give −1/x².

Writing (uv)′ = u′v′.

(uv)′ = u′v + uv′. For x · x, u′v′ gives 1 but the true answer is 2x.

Finding lim (x¹⁵ − 1)/(x¹⁰ − 1) as 15 · 10 or as 1.

Divide top and bottom by x − 1 and apply naⁿ⁻¹ to each: 15/10 = 3/2.

7 to know

Average velocity

v = [s(t₂) − s(t₁)]/(t₂ − t₁)

Slope of the chord of the distance-time graph.

Standard power limit

lim_{x→a} (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹

Any positive integer n; also rational n when a > 0.

Trigonometric limits

lim_{x→0} (sin x)/x = 1, lim_{x→0} (1 − cos x)/x = 0

x in radians.

Derivative at a point

f′(a) = lim_{h→0} [f(a + h) − f(a)]/h

Slope of the tangent at (a, f(a)).

Product and quotient rules

(uv)′ = u′v + uv′, (u/v)′ = (u′v − uv′)/v²

v ≠ 0 in the quotient.

Power rule

d/dx (xⁿ) = nxⁿ⁻¹

Constant: derivative 0.

Trigonometric derivatives

(sin x)′ = cos x, (cos x)′ = −sin x, (tan x)′ = sec² x

(cot x)′ = −cosec² x.

8 terms

Average velocity: Distance covered in a time interval divided by the length of the interval.
Instantaneous velocity: The limit of average velocities over intervals shrinking to one instant.
Limit: The value f(x) settles towards as x approaches a, whatever f(a) is.
Left and right hand limits: Limits found as x approaches a from below and from above.
Sandwich theorem: A function squeezed between two functions with the same limit has that limit too.
Derivative: The limit of [f(x + h) − f(x)]/h as h → 0; the rate of change of f.
First principles: Finding a derivative directly from its limit definition.
Tangent: The limiting position of a chord through P as its other end slides to P.