Derivatives
0
∫1/x^2dx
-x^-1 + c
What is the definition of Intermediate Value Theorem (IVT)
If the function f(x) is continuous on the closed interval [a,b] and f(b) < c < f(a) then there must exist some value c on that interval
When f'(x) > 0 what is f(x) doing
f(x) is increasing
What is the derivative of cos(x)
-sin(x)
What is the derivative of 3x^2
6x
∫(2x-3x^2)dx
x^2 - x^3 + c
What is the definition of Extreme Value Theorem (EVT)
If f(x) is continuous on the close interval [a,b] then there must be at least one maximum or one minimum
When f'(x) < 0 what is f(x) doing
f(x) is decreasing
What is the derivative of tangent
sec^2(x)
What is the derivative of (x^2)((sin(x))
x(2sin(x) + xcos(x), using product rule
∫(√x + 1/2√x) dx
2x^3/2 /3 + x^1/2 +c
What is the definition of Mean Value Theorem (MVT)
If the function f(x) is continuous on [a,b] and differentiable on (a,b) then there must be some point where f'(c) = f(b)-f(a)/b-a
When f'(x) = 0 what does f(x) have
f(x) has critical points
-csc^2(x)
What is the derivative of 7x+4/x^2 + 5
-7x^2 - 8x + 35/ (x^2 + 5)^2, using quotient rule
∫(sin(2x)+cos(2x))dx
-1cos(2x)/2 + 1 sin(2x)/2 + c ,by u substitution or pattern recognition
What is the fundamental Theorem of Calculus
if f(x) is continuous on [a,b] and differentiable (a,b) then the integral from a to b of f(x)dx = f(b) - f(a)
When f''(x) = 0 and changes sign what does f(x) have
f(x) has point of inflections
What is the derivative of arcsin(x) or sin⁻¹(x)
1/√1-x^2
What is the limit definition of the derivative
limit as h approaches 0 of f(x + h) - f(x)/ h
∫x/x^2 - 4dx
1ln[x^2 - 4]/2 + c
What is the second fundamental theorem of Calculus
The derivative of integral from a to x f(t)dt = f(x)
How do you find the absolute minimum or maximum values for a function when given f'(x)
You can use a t chart with your end points and include your critical points and then plug those x values to your original function or you can use candidates test
If f(x) = cos(3x), then f'(π/9)
-3√3/2