Power Rule
Derive: x2
2x
Let g be a function such that g(4)=8, and g'(4)=-3
Let h be the function h(x)=x1/2
Let H be a function defined as H(x)=g(x)⋅h(x)
What is H′(4)=?
-4
Derive: sin(x)
cos(x)
Derive: sin(x2)
2xcos(x2)
For y=x, what is the limit as x approaches 1.
1
Derive: x3
3x2
Table:
x g(x) h(x) g'(x) h'(x)
0 -3 -1 5 3
F(x)=g(x)⋅h(x)
What is F'(0)=?
-14
Derive: cos(x)
-sin(x)
Derive: (x2)1/2
2x/(x2)1/2
Approximate the area between the x-axis and f(x) from x=10 to x=16 using trapezoidal sum with 3 equal subdivisions:
x 10 12 14 16
f(x) 5 1 7 7
28
Derive: x10
10x9
Derive: cos(x)/sin(x)
-1/sin(x)2
Derive: sec(x)
(sec(x))(tan(x))
Table:
x f(x) g(x) f'(x) g'(x)
1 4 2 3 -2
2 6 1 1 0
G(x)=f(g(x)), what is G'(2)?
G'(2)=0
A particle moves in a straight line with a velocity v(t) meters per second, where t is the time in seconds. At t=3, the particle's distance from the starting point was 12 meters in the positive direction. What was the particle's position at t=9 seconds?:
How would you write an expression to represent this?
12 + 93| v(t) dt
Derive: 3x12
36x11
Table:
x g(x) h(x) g'(x) h'(x)
-2 4 1 -1 2
H(x)=g(x)/h(x)
What is H'(-2)=?
H'(-2)=-9
Derive: arcsin(x)
1/(1-x2)1/2
f(x)= -5 x 7x
Find f '(x)
-5 x 7x ln(7)
Let f be a twice differentiable function, and let f(-7)=6, f'(-7)=0, and f''(-7)=-5
What occurs in the graph of f at the point (-7,6)?
(-7,6) is a maximum point
Derive: 9x6
54x5
Derive: y=x2 ln(x)
x+2xln(x)
Derive: cot(x)
-csc2(x) or -1/sin(x)2
y= tan(x2 - 4x)
Find dy/dx (Derivative)
2x - 4 / cos2(x2 - 4x)
f(x)=2x3-9x2+12x
Where does f have critical points?
x= 1 and 2