What is the second derivative of f(x) =x4
4x3
12x3
12x2
24x
1/x
±1/x
-x/1
none
What does the first derivative f’(x) represent geometrically?
The area under the curve
The slope of the tangent line to the curve
The average value of the function
The rate of change of the integral
What does the critical point of a function represent geometrically?
A point where f’(x) is maximum
A point where f’(x)=0 or f’(x) does not exist
A point of inflection
A point where f(x) equals to 0
What does the second derivative f’’(x) represent geometrically?
The curvature or concavity of f(x)
The rate of change of the tangent line’s slope
The acceleration of f(x) (if interpreted as motion)
All the above
What does represent geometrically?
The slope of the tangent line at x = b
The area under f(x) from x = a to x = b
The net change in f(x) from x = a to x = b
The total distance traveled along f(x)
Why is the derivative of a constant c,f’(x)=0
Because the slope of the tangent to y = c is zero everywhere
Because the rate of change of c is zero
Because c does not depend on x
All the above
If =0 which of the following is true
f(x) must be 0 everywhere in [a, b]
F(x) must be positive in [a, b]
f(x) could have equal positive and negative areas in [a, b]
F(x) must have no critical points
If , what does y present?
An linear function
A constant function
An exponential function
A quadratic function
Compute
1/4
1/2
1
0
Let f(x)=x3-6x2+9x+4 Find the local max and min.
1 and 3
2
1 and 2
2 and 3
If f(x) =xx,what is f’(x)?
Xx(1+ln(x))
(X-1)x
xxlnx
none
If y=arcsinx, what is
none
Evaluate
not exist
0
∞
1
not exist
0
1
∞