(1) Engineering Mathematics,. K.A. Stroud (D. Booth),. Palgrave, 5th. edition,
2001. References to chapters will be given regularly. (2) Mathematics for
Engineers ...
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1M1 Mathematics Mechanical, Civil and Aerospace Engineering Introduction Lecturer: Dave Harris email:
[email protected] Office: MSS/N12b Course coordinator: Dr. C. Steele email:
[email protected] Office: MSS/N7 Websites: (1) Service course website http://www.ma.man.ac.uk/servicemaths/ (2) www.maths.manchester.ac.uk/ then follow links to and from Dave Harris’s website: http://personalpages.umist.ac.uk/staff /david.harris/
Examples sheets etc.: lecture or website. Information about tutorial groups: see servicemaths website. Books: (1) Engineering Mathematics, K.A. Stroud (D. Booth), Palgrave, 5th. edition, 2001 References to chapters will be given regularly. (2) Mathematics for Engineers, A. Croft & R. Davidson, Prentice Hall, 1999
1.Functions 1.1 Definition Given two sets of numbers, A, B, a function f is a rule which given a number x in A generates a new number y = f (x) in B. y = f (x) x is called the argument. 1
y is called the value. x is the independent variable. y the dependent variable. 1.2 Definition Given a function f , the inverse function f −1 (if it exists) is the function such that if y = f (x) then x = f −1 (y) .
Powers, indices, exponents
1.2 Definition: Given a number a and any positive integer n, (i) an
=
a × a × ... × a | {z } ntimes
a is raised to the power n. n is the power, index or exponent. (ii) Negative power a−n =
1 an
(ii) Zeroth power: √ (iv) Fractional powers: a1/n = n a 1.3 Theorem: For positive integers m, n (i) an × am = an+m (multiply powers: add indices) (ii) an ÷ am = an−m (divide powers: subtract indices) n (iii) (am ) = amn (raise a power to a power: multiply indices). 1.4 Theorem: Laws of indices for any numbers a, x, y
a0 = 1
ax ay = ax+y y
(ax ) = axy a−x =
1 ax
1.5 Definition: Let a > 0, a 6= 1. If ab = c then write b = loga c. Say “b is the logarithm of c to the base a. Thus, b = loga c is the power (exponent) to which a must be raised to give c. 1.6 Theorem: (i) loga 1 = 0. (ii) loga a = 1. 1.7 Theorem: Laws of logarithms a > 0 is the logarithm base, u, v, w are positive numbers 2
(i) loga (uv) = loga u + loga v, (log of a product = sum of the logs) (ii) loga (u/v) = loga u − loga v, (log of a quotient = difference of the logs) (iii) loga (uw ) = w loga u, (log of a power = power times the log.) 1.8 Theorem: Change of base a, b are two bases, a, b > 0, u is a positive number logb u =
loga u . loga b
1.9. Definition: The exponential function is the number e raised to the variable power x. 1.10. Definition: The natural logarithm function y = ln x is the function such that x = exp y.
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Polynomials
1.11 Definition: A polynomial in variable x, P (x) is a finite sum of terms, each term a non-negative whole power of x multiplied by a constant. Highest power in P (x) is the degree of P (x) . 1.12 Definition: Completing the square Bring a quadratic P (x) = ax2 + bx + c where a 6= 0, to the form 2
y = A (x + B) + C, where A, B, C are constants. Stage 1: Write
µ ¶ b P (x) = a x2 + x + c a
Stage 2: Note b x + x= a 2
µ ¶2 b b2 x+ − 2 2a 4a
Stage 3: Substitute 2 into 1: "µ P (x) = a
b x+ 2a 3
¶2
# b2 − 2 +c 4a
Stage 4: Simplify
µ ¶2 b b2 P (x) = a x + +c− 2a 4a
1.15 Theorem:
N (x) R (x) = Q (x) + D (x) D (x)
where polynomials Q (x), R (x) are called the quotient, remainder respectively, and degree R < degree D.
2.1
Binomial theorem
Expression for (a + b)n 0
(a + b) 1 (a + b) 2 (a + b) 3 (a + b) 4 (a + b)
= = = = =
1 a+b a2 + 2ab + b2 a3 + 3a2 b + 3a2 b + b3 a4 + 4a3 b + 6a2 b2 + 4ab3 + b4
1.16 Theorem: n
= an + nan−1 b +
(a + b)
... +
n (n − 1) n−2 2 a b + 1.2
n (n − 1) 2 n−2 a b + nabn−1 + bn 1.2
Pascal’s Triangle for Binomial coefficients 1 1 1 1 1 1 1
1
3 4
5 6
1 2 3 6
10 15
1 4
10 20
1 5
15
1.17 Definition: n!
= n (n − 1) (n − 2) ...2.1
0!
=
n
1 µ
Ck =
n k
1.16a Theorem: n
(a + b) =
¶ = n X k=0
4
n! k! (n − k)!
.n Ck an−k bk
1 6
1
Example 1: µ
999 1000
¶25
¶25 1 1000 µ ¶ 1 = 1 + 25 × − 1000 µ ¶2 1 25.24 × − + ... + 2 1000 25 975 ≈ 1− = 1000 1000 =
µ 1−
Example 2 7
(1 + x)
7.6 2 7.6.5 3 7.6.5.4 4 x + x + x 1.2 1.2.3 1.2.3.4 7.6.5.4.3 5 7.6.5.4.3.2 6 + x + x + x7 1.2.3.4.5 1.2.3.4.5.6 = 1 + 7x + 21x2 + 35x3 +35x4 + 21x5 + 7x6 + x7
= 1 + 7x +
Similarly (2 + x)
2.2
6
26 + 6.25 x + 15.24 x2 + 20.23 x3 +15.22 x4 + 6.2x5 + x6
=
Trigonometry
Right angled triangle 1.18 Definition: sin θ
=
cos θ
=
tan θ
=
opposite o = , hypotenuse h adjacent a = , hypotenuse h sin θ opposite o = = cos θ adjacent a
for 0◦ < θ < 90◦ (0 < θ < π/2radians). Extend to other angles using sum/difference identities (see handout). 1.19 Definition: csc θ
=
cot θ
=
1 , sin θ 1 . tan θ
sec θ =
5
1 , cos θ
Pythagoras’ theorem:
a2 + o2 = h2
gives trig. identities a2 o2 + 2 = 1 : cos2 θ + sin2 θ = 1 2 h h o2 h2 = : 1 + tan2 θ = sec2 θ a2 a2 h2 a2 + 1 = : cot2 θ + 1 = csc2 θ o2 o2
1+
2.2.1
Radians and degrees
Proportion of circle occupied by arc is x θ or 360 2π If x radians and θ degrees represent same angle then θ x = 360 2π Also, for radius r, θ radians, arc s s = rθ. 2.2.2
Inverse trigonometric functions
Inverse sine function x = sin−1 y ⇔ y = sin x x is the angle, − 12 π < x < 12 π, whose sine is y. Inverse cosine function x = cos−1 y ⇔ y = cos x x is the angle, 0 < x < π, whose cosine is y. Inverse tangent function x = tan−1 y ⇔ y = tan x x is the angle, − 12 π < x < 12 π, whose tangent is y. Write y = sin−1 x y = cos−1 x y = tan−1 x
Beware!
−1
means inverse not reciprocal. 6
2.2.3
Inverse Functions
Recall defn. 1.2. If f and g are functions and y = f (x) ⇒ x = g (y) then g is the inverse function of f . Write g = f −1 . so
y = f (x) ⇒ x = f −1 (y)
Beware! f −1 (x) 6= usually. Note also
1 f (x)
¡ ¢ y = f f −1 (y) x = f −1 (f (x))
Examples: Pairs of mutually inverse functions √ y = x2 y=+ x y = exp x y = ln x y = sin x y = sin−1 x y = cos x y = cos−1 x y = tan x y = tan−1 x To plot graph of f −1 (x) : reflect graph of y = f (x) in line y = x.
2.2.4
Points and lines
Cartesian coordinates A line is characterised by • slope (gradient) m one point on the line (e.g. intercept with y-axis (x = 0), c • two distinct points on line Distance R between two points 2
2
2
2
R2 = (x1 − x2 ) + (y1 − y2 ) (Pythagoras’ theorem) so q R=
(x1 − x2 ) + (y1 − y2 ) . 7
Equation of a line y = mx + c Gradient =
change in y . change in x
Given two points on line (x1 , y1 ) and (x2 , y2 ) Gradient =
y1 − y2 (= m) . x1 − x2
If (x, y) is any point on the line Gradient =
y − y2 (= m) . x − x2
Equating the two expressions y − y2 y1 − y2 = . x − x2 x1 − x2 This is the equation of the line.
2.2.5
Polar coordinates
The point P (x, y) can be found by specifying distance, r, from origin O (0, 0), and angle that OP makes with the +ve x-axis, measured anti-clockwise. tan θ : gradient of line through OP r : length of OP So y tan θ = x r2 = x2 + y 2 or θ r
³y´ = tan−1 x p 2 = x + y2
Bewarequadrant!
Given polar coordinates (r, θ) cos θ
=
sin θ
=
x r y r
i.e. x = y =
r cos θ r sin θ 8
gives the Cartesian coordinates. Note: sin and sin−1 etc. are better defined for angles in radians. 1 −1 = csc x = (sin x) sin x Correction:
a2 h2 + 1 = : cot2 θ + 1 = csc2 θ o2 o2 Example 1:√ Find ³ ´ the polar coordinates of the point P with Cartesian coordinates 21 , − 23 . v uµ ¶2 Ã √ !2 u 1 3 + − = 1. r=t 2 2
tan θ
=
θ
=
− −
√ 3 2 1 2
√ =− 3
π 3
Example 2: Find the inverse function of f (x) = 4x + 36. Let y = 4x + 36 then x = f (y) is given by x=
1 1 (y − 36) = y − 9 4 4
So f −1 (y) =
1 y−9 4
or y = f −1 (x) =
1 x−9 4
Example 3: Try for yourselves: if f (x) =
1 +3 x
find f −1 (x) . Example 4: Find the equation of the line through (1, 1) and (4, 5) . y−1 5−1 4 = = x−1 4−1 3 So y−1= or y=
4 (x − 1) 3
4 1 x− 3 3 9
2.2.6
Intersection of two lines
Given two lines L1 L2
y y
= m1 x + c1 = m2 x + c2
(1) (2)
Point of intersection: (simultaneous linear equations): (1)-(2) 0 = (m1 − m2 ) x + (c1 − c2 ) If m1 6= m2 (lines not parallel) x=−
c1 − c2 m1 − m2
and y
c1 − c2 + c1 m1 − m2 m1 c2 − m2 c1 m1 − m2
= −m1 =
(unique solution) If m1 = m2 (lines parallel) • c1 = c2 : same line (infinitely many solutions) • c1 6= c2 : different parallel lines (no solution) 2.2.7
Circles
Circle centred at (a, b), radius R : Square of distance of point (x, y) to (a, b) 2
2
(x − a) + (y − b) = R2 Example 5:
x2 + 2x + y 2 = 8
is the equation of a circle, centre: (−1, 0) , radius 3 units. Completing the square (on x, y separately) 2
(x + 1) − 1 + y 2 2
(x + 1) + y
10
2
= 8 = 32
2.2.8
Parametric equations
If y = f (x) then the point (x, f (x)) lies on graph. ¡ ¢ Example 6: For the parabola y = x2 , the point x, x2 lies on it. ¡ ¢ Let x = t, then y = x2 = t2 so the curve consists of exactly the points t, t2 as the real variable t varies. This is called a parametric form of the curve. In general, y = f (x) may be described by x = x (t) ,
y = y (t) .
By stating the values of t allowed, a part of the curve may be specified. Example 7: If x = t2 and y = t4 (0 ≤ t ≤ 1) find y as a function of x and sketch the graph. Solution: x2 = t4 = y, (0 ≤ x ≤ 1) y = x2 .
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