Limits
the limit as x approaches -5 of
(2x+10)/(x2+2x-15)
-1/4
Find the derivative of:
f(x)= pi^2 +e^3 -4
0
If x2+xy-y=2, find dy/dx
(-2x-y)/(x-1)
or
(2x+y)/(1-x)
The following are two conditions for continuity:
A. f(a) is defined
B. the limit as x approaches a of f(x) exists
What is the third condition?
What is the limit as x approaches a of f(x) equals f(a)
The definition of a derivative in limit notation
What is the limit as x ->a of (f(x)-f(a))/(x-a)
or
the limit as h->0 of (f(x+h)-f(x))/h
If y=(x3+1)2, then dy/dx=
6x2(x3+1)
If f(x)=sin2(3-x) then f'(0)=? Your answer can be left in terms of sin and cos.
-2sin3cos3
If the graph of f'''(x) of some function f is a line of slope 2, then f'(x) could be what type of function?
A cubic function
True/False: If f(a) is undefined, then the limit as x approaches a will also be undefined.
False: Not Necessarily
If f(x)=cos(3x), then f'(pi/3)=
0
The circumference of a circle is increasing at a rate of (2pi)/5 inches per minute. When the radius is 5 inches, how fast is the area of the circle increasing in square inches per minute?
2pi
The Mean Value Theorem (state it)
A function that is continuous and differentiable on (a,b) then there exists a number c such that
f'(c)=(f(b)-f(a))/(b-a)
If the limit as x approaches infinity of
6x2/(200-kx-kx2) = 2/3, then k=
-9
If f(9)=3 and f'(9)=-2, find the derivative of x2f(x2) at x=3
-90
A conical tank has a height that is always 3 times its radius. If water is leaving the tank at a rate of 50 cubic feet per minute, how fast is the water level falling in feet per minute when the water is 3 feet high?
v=(1/3)pi r^2 h0
50/pi or about 15.915
f(x)={x+2 if x is less than or equal to 3; 4x-7 if x>3}
Which are true?
I. the limit as x approaches 3 of f(x) exists
II. f is continuous at x=3
III. f is differentiable at x=3
I and II