Limits
Limit as x approaches 0 of sin(x)
sin(0) = 0
Integral of x4
(x5/5) + C
Derivative of cos(x)
-sin(x)
Where is f(x) = 2/(x-3) continuous?
Where x < 3 and x > 3 or
(negative infinity, 3) and (3, positive infinity)
Find absolute minimum for f(x) = x2-4x on interval (negative infinity, infinity)
derivative is 2x-4
2x - 4 =0
x = 2
(2)2-4(2) = -4
Minimum is (2, -4)
Limit as x approaches 2 of (x2-4)/(x-2)
(2) + 2 = 4
Integral of x3+4
(x4/4) + 4x + C
Derivative of tan(x)
sec2(x)
Where is f(x) = (4x2-36x) / (x-9) discontinuous?
x = 9
Does f(x) = 4x2+2x+3 have a maximum and/or a minimum?
Polynomial opening up, so it has a minimum but no maximum
Limit as x approaches infinity of (2x4 - x2 - 8x)
Integral of exx
exx - ex + C
Derivative of cot(x)
-csc2(x)
Where is f(x) = |x|/x discontinuous?
x = 0
Find absolute max and min of f(x) = -(x+4)3 on interval [-4, -3)
Absolute max at (-4, 0), no absolute min
Limit as x approaches 0+ of (1/x)
Infinity
Integral of sin(2x)
-(1/2)cos(2x) + C
Derivative of sec(x)
tan(x)sec(x)
Where is f(x) = (4-x2)1/2 continuous?
-2 </= x </= 2 or
(negative infinity, -2] and [2, infinity)
The daily profit, P, of an oil refinery is given by P = 8x-0.02x2 where x is the number of barrels refined. How many barrels will give the max profit?
dP/dx has to be 0
0 = 8-0.04x
x = 200 barrels
Limit as x approaches 1- of x/(x-1)
Negative infinity
Integral of x/(x2+1)
(1/2) ln |(x2+1)| + C
Derivative of csc(x)
-cot(x)csc(x)
f(3) = 7, so continuous
A rectangular storage area is to constructed along the side of a tall building. A fence is required along the remaining 3 sides of area. What is the max area that can be enclosed with 800 m of fencing?
Area is A=xy
2x + y = 800, y = 800-2x
A = x(800-2x) = 800x-2x2
dA/dx = 800-2x2 = 0
x = 200
2(200) + y = 800
y = 400
A = (200)(400) = 80 000 m2