Functions
Is the following function even or odd?
f(x) = x/((-x^2)-1)
odd
- substitute -x and if f(-x)=-f(x) then it is odd
Evaluate:
the limit as x approaches -1 of (x^2 +6x+5)/(x^2 -3x-4)
-4/5
Find the second derivative of the following function:
f(x) = cos(x) - sin(x)
f"(x) = sin(x) - cos(x)
Find the antiderivative of the following function
x^4 + 4x^3 dx
((x^5)/5) + x^4 +C
solve for x
e^x = 1/9
x = ln(1/9)
Find the domain of the function
f(t) = sqrt(t-1) + sqrt(t+6)
domain = [1, inf)
does the following limit of g(x) as x approaches 2 exist
g(x) = { (x^2 +4x -12)/(x^2 - 2x) if x does not equal 2
{ 6 if x=2
yes
Use logarithmic differentiation to find dy/dx of
y = x^(sinx)
dy/dx = x^(sinx) (cosxlnx+(sinx/x))
Evaluate the indefinite integral
sinx + 10csc^2(x) dx
-cos(x)-10cot(x)+c
find the horizontal & vertical asymptotes of f(x)
f(x) = 1/(x-1)
horizontal = 0
vertical = 1
Simplify the following:
1. ln(cos(theta)) - ln(cos(theta)/2) (150pts)
2. e^(-ln(x^2)) (150pts)
1. ln(2)
2. 1/(x^2)
evaluate:
1. limit as x approaches infinity of (2x+3)/(5x+7)
2. limit as x approaches -infinity of (x+67)/(x^3 +3)
1. 2/5
2. 0
Find the line that is tangent to the curve at the point (0,3)
(y^2)(e^(2x))=3y+(x^2)
tangent line is y=-6x+3
Evaluate the integral
10sin(2x)cos(2x)sqrt(cos^2(2x)-5)
-(5/3)((cos^2(2x)-5))^3/2 +c
A thin sheet of ice is in the form of a circle. If the ice is melting in such a way that the area of the sheet is decreasing at a rate of 0.5 m2/sec at what rate is the radius decreasing when the area of the sheet is 12 m2?
r' = -0.040717
Find the inverse of the function:
f(x) = (x^3)+6
f^-1(x) = cubed root of (x-6)
Is the function continuous at
1. x=4
2. x=6
1. yes
2. no because lim as x approaches 6 of g(x) does not exist
A person is standing 350 feet away from a model rocket that is fired straight up into the air at a rate of 15 ft/sec. At what rate is the distance between the person and the rocket increasing 20 seconds after lift off?
9.76187
Evaluate the Integral with lower bound of 0 and upper bound of pi/2
7sin(x)-2cos(x) dx
5
Find the area of the region bounded by the following
y=x^2 +2 (above)
y= sinx (below)
x=-1
x=2
8.04355
let f(x) = sqrt(3x-2)
1. find inverse
2. find the domain of the answer to #1
1. ((x^2)+2)/3
2. domain = (-inf,inf)
find the derivative of the function using the definition of a derivative
f(x) = (x+1) / (x+4)
f'(x) = 3 / (t+4)^2
We have a piece of cardboard that is 50 cm by 20 cm and we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will give a maximum volume.
h = 4.4018cm
Evaluate the integral with lower bound of pi and upper bound of 0
sinxcos^3(x)dx
0
Let f(x)= (x^4) - (2x^3)-(12x^2)
Find
1. Find local min, max and inflection point (300 points)
2. Increasing/Decreasing intervals (100 points)
3. Concave up vs concave down intervals (100 points)
1. local max = 0, local mins = -1.8117 & 3.3117, inflection points (-1,-9) & (2,-48)
2. increasing: (-1.8117,0) & (3.3117, inf) decreasing: (-inf, -1.8117)
3. concave up: (-inf, -1) & (2,inf) concave down: (-1,2)