Derivatives
Integrals
Area & Volume
Graphs
Miscellaneous
100
Name 3 of the 6 derivative rules
Power Rule Product Rule Quotient Rule Chain Rule Constant Rule Sum/Difference Rule
100
What does an integral represent?
Area under a curve
100
What does the definite integral from a to b of f(x) represent?
The area above the x-axis - the area below the x-axis
100
If F'' is positive, what is F? If F'' is negative?
F is concave up. F is concave down
100
How do you find the vertical asymptotes of a function?
Do all factor/cancel of f(x) and set the denominator of f(x) equal to zero
200
What is the definition of a derivative?
lim (f(x+h) - f(x))/h h→0
200
What is the antiderivative of a constant? For example what is the antiderivative of: ∫K du
∫K du = Ku + C
200
What is the method that uses ⫪∫(R^2-r^2)?
Washer Method
200
When does F' have a relative min? Relative max?
F'' changes from negative to positive. F'' changes from positive to negative
200
What are the three conditions for f(x) to be continuous?
1) lim f(x) exists x→a 2) f(a) exists 3) lim f(x) = f(a) x→a
300
What are the derivatives of the 6 trigonometric functions?
d/dx (sin u) = cos(u) • u' d/dx (cos u) = -sin(u) • u' d/dx (tan u) = sec^2(u) • u' d/dx (sec u) = sec(u)tan(u) • u' d/dx (cot u) = -csc^2(u) • u' d/dx (csc u) = -csc(u)cot(u) • u'
300
What are the antiderivatives of the 6 trigonometric functions?
∫cos(u) du = sin(u) + C ∫sin(u) du = -cos(u) + C ∫sec^2(u) du = tan(u) + C ∫sec(u)tan(u) du = sec(u) + C ∫csc^2(u) du = -cot(u) + C ∫csc(u)cot(u) du = -csc(u) + C
300
How do you find the area between two graphs?
A = ∫ Ytop - Ybottom dx A = ∫ Xright - Xleft dy
300
Where does a function have relative extrema?
There is a relative minimum where the derivative of the function changes from negative to positive. There is a relative maximum where the derivative of the function changes from positive to negative.
300
How do you approximate the value of f(0.1) by using the tangent line to f at x=0?
Find the equation of the tangent line to f using y-y1=m(x-x1) where x1=0, y1=f(0) and m=f'(0). Then plug in 0.1 into x of the equation of this line. Be sure to use an approximate sign (≈)
400
What are the derivatives of the following inverse trigonometric functions? sin^-1(u) tan^-1(u) sec^-1(u)
d/dx (sin^-1(u)) = u'/√(1 - u^2) d/dx (tan^-1(u)) = u'/(1 + u^2) d/dx (sec^-1(u)) = u'/(|u|√(u^2 - 1))
400
What are the antiderivatives of the following inverse trigonometric functions? ∫du/√(1 - u^2) ∫du/(1 + u^2) ∫du/u√(u^2 - 1)
∫du/√(1 - u^2) = sin^-1(u) + C ∫du/(1 + u^2) = tan^-1(u) + C ∫du/u√(u^2 - 1) = sec^-1(u) + C
400
What are the area formulas of the basic shapes? (Circle, Semicirlce, Square, Isosceles Triangle, and Equilateral Triangle)
Circle: A = ⫪r^2 Semicircle: A = (1/2) ⫪r^2 Square: A = s^2 Isosceles Triangle: A = (hyp)^2/4 Equilateral Triangle: A = (√3/4)s^2
400
How do you find the average rate of change? Average value?
Average rate of change: (f(b) - f(a))/(b - a) Average value: ∫f(x) dx/(b - a)
400
What does SVAT represent? What is the relationship between each letter?
S represents position. V represents velocity. A represents acceleration. There are all functions with respect to time, T. The derivative of S is V and the derivative of V is A. The antiderivative of A is V and the antiderivative of V is S
500
What is the derivative of the following function? f(x) = -sec(ln(e^3/2)) - sin^-1(tan(2x))
f'(x) = 2sec^2(2x)/√(1 - tan^2(2x))
500
What is the antiderivative of the following function? ∫x^3/(1 + x^4)^1/3 dx
∫x^3/(1 + x^4)^1/3 dx = (3/8)(1 + x^4)^2/3 + C
500
The base of an object is determined by the two curves y=x^2/10 and y=-x^2/10 for 1 ⩽ x ⩽ 4. For this object, the cross sections perpendicular to the x-axis are squares. What is the volume of this object?
V = 8.184
500
How do you find points of inflection from the graph of f'?
Look for any relative extrema
500
What are the two steps to solving a differential equation?
Separate and Integrate
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