SOME CALCULUS STEP-BY-STEP PLANS, 2WBB0
c MICHIEL HOCHSTENBACH, FALL 2022
Here an overview of some step-by-step plans from the slides. Try to learn these by heart by
following them to solve exercises; work systematically.
Zeros of 3rd and higher-degree polynomials p(x) = 0
1.
2.
3.
4.
5.
First check if it is an easier special case, e.g. x3 − 8 = 0 or x4 + 2x2 + 1 = 0
If not, guess a zero, say x = a
Divide: p(x)/(x − a) (should have rest 0), this gives a new polynomial q(x)
If the degree of q(x) is still > 2, repeat this procedure
If the degree of q(x) equals 2, try sum–product, or else quadratic (“abc”) formula
Equations / inequalities with rational functions
1. Equations: collect everything into 1 fraction, or multiply crosswise
2. Inequalities: collect everything into 1 fraction with “> 0” or “< 0” on the right-hand
side. Then make a table of signs of nominator and denominator; together this determines
the signs of the rational function.
Equations / inequalities with sin, cos, tan
1. Replace everything as much as possible by 1 type of function, you can use e.g.:
sin2 (x) → 1 − cos2 (x) or vice versa
sin(x)
tan(x) → cos(x)
sin(2x) → 2 sin(x) cos(x)
cos( π2 − x) → sin(x) etc
Goal is to get an equation or inequality in only 1 unknown; avoid square roots.
Avoid having both sin(x) and cos(x), or sin(x) and sin(2x)
2. Carry out a substitution, e.g. y = sin(x), and solve the equation / inequality.
This can contain a polynomial or rational function.
3. Replace y again by by e.g. sin(x), and solve, e.g. sin(x) = 21 ⇒ x = π6 or x = − π6
4. Think of “+k 2π” (for sin and cos) or “+k π” (for tan)
Equations / inequalities with square roots
1. Equations: solve by squaring (possibly several times); always check solutions at the end.
2. Inequalities: first solve equality; for the solution also take care that all arguments under
square roots should be ≥ 0.
1
Equations / inequalities with powers
1. Write both sides as power of the same basis
2. Then omit the basis; an inequality sign flips when the basis is between 0 and 1
3. Alternative for powers of basis e: we can also take the ln
Equations / inequalities with ln
1.
2.
3.
4.
Write both sides as ln (without something in front)
Omit the ln’s
Take the domains of the ln’s into account
Alternative: you can also eliminate the ln’s by taking e-powers
Differentiating f (x)g(x)
1. Write f (x)g(x) as eg(x) ln(f (x))
2. Take the derivative: eg(x) ln(f (x)) times the derivative of the exponent
3. Write back eg(x) ln(f (x)) to f (x)g(x)
Optimization: finding critical points of f (x) (for minima/maxima)
1.
2.
3.
4.
5.
6.
7.
Solve f 0 (x) = 0
Make a sign scheme of f 0
Use this to determine: local max, local min, or reflection point
Consider the boundary points if the domain is bounded
Consider possible points where f 0 does not exist (“cusps”)
Determine the function value in all these points
Compare the local max/min’s to determine the global min/max
Limits
1. Always first try to fill in the number
if this gives no problem (see items 2–8): done!
2. 6=00 : consider lim and lim (ONLY for this type!)
x→a+
3.
4.
5.
6.
7.
8.
NB:
x→a−
• If ∞ and ∞ then limit ∞ (some say: does not exist)
• If −∞ and −∞ then limit −∞ (some say: does not exist)
• If ∞ and −∞, or −∞ and ∞ then limit does not exist
0
:
0 l’Hôpital (possibly several times), or Taylor, or divide out factor
∞
∞ : divide by largest in denominator (or l’Hôpital)
oscillate
∞ : is 0, use Squeeze Theorem
∞ − ∞: write as 1 term
0 · ∞: write as 00 of ∞
∞
∞0 , 1∞ , 00 : trick with eln
0
∞,
0∞ and ∞∞ are not problematic
2
Taylor series in a = 0
1. Use a standard series, which you can modify
2. Or use the definition: f (0) + f 0 (0) x + 12 f 00 (0) x2 + 16 f 000 (0) x3 + · · ·
Taylor series in a 6= 0
1.
2.
3.
4.
5.
We want powers of y = x − a
Write x = y + a and substitute this into f (x)
Try to get a Taylor series with powers of y (and not, e.g., y − 1)
Replace y by x − a
Alternative: use the definition: f (a)+(x−a)f 0 (a)+ 12 f 00 (a) (x−a)2 + 61 f 000 (a) (x−a)3 +· · ·
Integration
1. Standard integrals
2. Substitution
• Recognize derivative
• u = the quantity of which you recognize the derivative
• If you can choose: substitute hardest factor
3. Rational functions
• Polynomial division if degree nominator ≥ degree denominator
• Factorize denominator into linear or possibly quadratic factors
• Partial fraction expansion: separate cases for linear, quadratic, multiple linear factors
• Integrate each term separately (linear factor gives ln, quadratic ln or arctan)
4. Partial integration
• Product of 2 functions: take hardest
=G
R
• 1 function: take f = 1 (e.g.:
arctan(x)
dx)
R 2 x
• May have to repeat (e.g.: x e dx)
R
• May do twice and arrive at the same (e.g.: ex sin(x) dx)
Differential equations (ODEs)
dy
1. Separable differential equations dx
= f (x) g(y)
dy
• Separation of variables: g(y) = f (x) dx
R dy
R
• Integrate both sides g(y)
= f (x) dx + C
• Solve: integrate and express y in x
dy
2. First-order linear ODE dx
= p(x) y + q(x)
dy
• First solve homogeneous part dx
= p(x) y by separation of variables
dy
(special case of separable dx = f (x) g(y)), this contains a constant C
• Variation of constant: replace C by C(x)
• Substitute into ODE with inhomogeneous term, we get a new ODE with C 0 (x), but
this one can be solved by ordinary integration since the right-hand side does not
contain C(x)
• Solve for C(x) and determine solution ODE
3
Vectors step-by-step plans:
Distance of point (p, q, r) to plane ax + by + cz = d
1. Find parametrization line through point perpendicular to plane:
(x, y, z) = (p, q, r) + λ (a, b, c)
2. Compute intersection point line with plane:
substitute x = p + λa, y = q + λb, z = r + λc into plane equation
3. Compute distance between (p, q, r) and intersection point
Distance of point to line in R3
1. Give equation of plane through point perpendicular to line;
ax + by + cz = d, with (a, b, c) the direction vector of the line
2. Compute d by substituting the point
3. Compute intersection point line with plane
4. Compute distance between point and intersection point
Area triangle (a, b, c), (d, e, f ), (g, h, i) in R3
1. Compute cross product of (d, e, f ) − (a, b, c) and (g, h, i) − (a, b, c)
2. Area = 21 · length of this vector
Area triangle (a, b), (c, d), (e, f ) in R2
1. Compute cross product of (c, d, 0) − (a, b, 0) and (e, f, 0) − (a, b, 0)
2. Area = 12 · length of this vector
NB: there are also other methods; these may be used as long as they give the correct answer.
Volume parallelepiped in R3
1. Compute 3 direction vectors a, b, c between the 4 points
2. Volume = |(a × b) · c|
Intersection line of two planes in R3
1. Find two points in the intersection, for instance by substituting x = 0 and solving the
2 × 2 system, and then by doing the same for y = 0
2. OR compute the direction vector of the line as the cross product of the two normal
vectors, and in addition 1 point in the intersection
Angle between two planes in R3
1. Compute the angle between the two normal vectors
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