Section 1
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y= constant
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Date created
Mar 1, 2020
Cards (77)
Section 1
(50 cards)
y= constant
y' = 0
f(x)= function f'(x)= inverse function f(f⁻¹(x)) = x
f'(f⁻¹(x))[f⁻¹(x)]' = 1 [f⁻¹(x)]' = 1/ f'(f⁻¹(x))
Speed
| u(t) |
y= e^x
dy/dx= e^x × d(x)/dx
y= cosx
dy/dx= -sinx × d(x)/dx
y= arccotu
Relationship between Differentiability and Continuity
Differentiability implies continuity
Extreme Value Theorem
If f is continuous on a closed interval [a,b] then f has both a minimum and a maximum on the interval
Instantaneous Rate of Change Slope of Tangent Line
Lim ∆x→0 f(x+∆x)-f(x)/∆x Lim h→0 f(x+h)-f(h)/h
Trig Limit (sinx)
Squeeze Thm Lim θ→0 sinθ/θ = 1
y= arctanu
y= arcsecu
Normal Line
Normal line is a line parallel to the tangent line at the point of tangency.
General Position Formula for Free-Falling Objects
x(t)= (1/2)gt²+Vt₀+S₀ Meters: g = -9.8m/sec² Feet: g = -32ft/sec² V₀= initial velocity S₀= initial height g= gravity
Maximum height of an object
1) v(t)= 0 Solve for t 2) Plug that t into position function
y= cotx
y'=-csc^2(x)
Chain Rule y= f(g(x))
y' = f'(g(x)) × g'(x)
3 reasons limits don't exist
1. They are unbounded: Lim x→0 1/x² 2. Right-hand limit ≠ Left-hand limit: Lim x→0 |x|/x 3. Oscillating Behavior: Lim x→0 sin(1/x)
Quotient Rule
y = f(x)/g(x) y'= g(x)f'(x) - f(x)g'(x)/ [g(x)]²
Power rule y=x^n
dy/dx = nxⁿ⁻¹ × d(x)/dx
Technique for Evaluating Limits
1) Direct Substitution 2) Factoring 3) Long/Synthetic Division 4) Expand 5) Common Denominators 6) Multiply by the Conjugate
y= cscx
y'=-cscxcotx
y= lnu
y'= 1/u • u'
Vertical Tangent
dy/dx doesn't exist
Acceleration
Derivative of velocity
3 reasons derivatives do not exist
1. Sharp Turn 2. Vertical Tangent 3. Discontinuity
Product Rule
y = f(x)g(x) y' = f(x)g'(x) + f'(x)g(x)
Average Rate of Change Slope of Secant
= Y₁-Y₂/X₁-X₂ f(x+∆x)-f(x)/∆x
Horizontal Tangent
dy/dx= 0
Speed Decreasing
r(t) and a(t) have different signs
Intermediate Value Theorem
If f is continuous on the closed interval [a,b]. f(a)≠f(b), and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c)=k
y=sinx
dy/dx= cosx × d(x)/dx
Continuity at a Point
Given that f(x) at x=a I. f(a) must exist II. Lim f(x) x→a exists III. Lim f(x) x→a = f(a)
Speed Increasing
v(t) and a(t) have the same sign
Critical Number
Let f be defined at c. If f'(c)= 0 or if f is not differentiable at c, then c is a critical number
a) particle at rest b) particle to right/up c) particle to left/down
a) v(t)= 0 b) v(t) > 0 c) v(t) < 0
vertical asymptote
Non-removable discontinuities
y= tanx
y'= sec^2(x)
Instantaneous Velocity (Velocity= Speed and Direction)
Derivative of the position function
y= a^u (Constant to a Variable Power)
y= lna × a^u × u'
Hits the ground
1) x(t)= 0 2) Plug that t into position function
y= arccosu
Average Function
The change in position/The change in time
Trig Limit (cosx)
Lim θ→0 1-cosθ/θ = 0
y= secx
y'= secxtanx
Unit Circle
y= arccscu
Trig Table
y= log∨a(u)
y'= (1/lna × U) × U'
y= arcsinu
Section 2
(27 cards)