Free QCE Mathematical Methods Flashcard Review
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State the law: log_a(mn)
log_a(mn) = log_a(m) + log_a(n)
State the law: log_a(m/n)
log_a(m/n) = log_a(m) − log_a(n)
State the law: log_a(mn)
log_a(mn) = n·log_a(m)
State the law: a^(log_a(x))
a^(log_a(x)) = x, for x > 0
State the law: log_a(ax)
log_a(ax) = x
Exact value: sin(0°), sin(30°), sin(45°), sin(60°), sin(90°)
0, 1/2, √2/2, √3/2, 1 respectively
Exact value: cos(0°), cos(30°), cos(45°), cos(60°), cos(90°)
1, √3/2, √2/2, 1/2, 0 respectively
Exact value: tan(0°), tan(30°), tan(45°), tan(60°)
0, 1/√3 (or √3/3), 1, √3 respectively
Derivative: d/dx[e^(f(x))]
f'(x)·e^(f(x))
Derivative: d/dx[ln(f(x))]
f'(x)/f(x), where f(x) > 0
Derivative: d/dx[sin(f(x))]
f'(x)·cos(f(x))
Derivative: d/dx[cos(f(x))]
−f'(x)·sin(f(x))
Derivative: d/dx[tan(f(x))]
f'(x)·sec2(f(x)), or f'(x)/cos2(f(x))
State the chain rule
dy/dx = (dy/du)·(du/dx)
State the product rule: d/dx[u(x)·v(x)]
u'(x)·v(x) + u(x)·v'(x)
State the quotient rule: d/dx[u(x)/v(x)]
[u'(x)·v(x) − u(x)·v'(x)] / [v(x)]2
Condition for stationary points
f'(x) = 0
Nature of stationary point: if f''(x) < 0
Local maximum
Nature of stationary point: if f''(x) > 0
Local minimum
Second derivative test: concavity when f''(x) > 0
Curve is concave up (concave)