Derivatives
5x2+6x-2
10x+6
2x3-3x
(2,6)
592
If a function is differentiable, the function must also be _______.
Continuous
4x2+8x
(1,4)
904.7787
10x4+4x3-7x2+9x-2
40x3+12x2-14x+9
4x5+4x3-7x2+9
(0,3)
531
The Extreme Value Theorem states that if a real-valued function f is continuous on the closed interval [a, b], then f must attain a _______ and a ______, each at least once.
minimum, maximum
x4-5x3+7x-2
(0,2)
-22.6195
3x2+sin(4x+3)
6x-4cos(4x+3)
8x9-6x7+4x3+8x2-9
(0,3)
42,444.45
Product rule for integration
d/dx (f(x)g(x))=f(x)g'(x)+g(x)f'(x)
5x4+7ex+9
(0,4)
9017.5462
4x-7+6x-2-e6x
−6e6x−(12/x^3)−(28/x^8)
3sin(4x2)-ln(4x2)
(π /4, π /2)
-1.3993
This theorem states that if a function is differentiable across the interval (a,b), there exists a point c at which f'(c)=(f(b)-f(a))/(b-a).
Mean Value Theorem
4cos(5x+2)-ln(2x2)
(1,3)
-33.1292
ln(5x4-3x2)+e3x
3e^3x+((20x3−6x)/(5x4−3x2))
6xcos(8x3)
(π /4, π /2)
0.1259
Rolle's Theorem states that if a function is differentiable on the interval (a,b), then there exists a number c that a<c<b ad f'(c)=______.
zero
sin(4x3)-3x2+7
(0,3)
-35.954