Extrema
Determine the intervals of concavity and classify the critical points of f(x) = 4x^3 -3x^2 - 36x + 5
Concavity: (-inf,1/4) up, (1/4,inf) down
Min @ x=2
Max @ x = -3/2
Find (f-1)`(8) for f(x) = x^3
1/12
Find d2y/dx2 for sqrt(x)*sqrt(y)=2
2y/x
A farmer needs to construct two adjoining rectangular pens of identical areas. If each pen is to have an area of 1200 square feet, what dimensions. will minimize the cost of fencing
x = 40 ft
y = 30 ft
Determine if the MVT applies to this function, if so find the c for which it is valid.
h(x) = 1/2 * X^3 -x^2 on [-2,2]
c = -2/3
Find the extrema of f(x) = (3x^2 +1)/(x-2) using the first derivative test
Max: x = (12- sqrt(156))/6
min: x =(12 + sqrt(156))/6
Find (f-1)`(pi/2) for f(x)=2tan^-1(x)
1
Find the points with vertical and horizontal tangent lines for (x^2 -2x + 5)*y = 5
Horizontal: (1,4/5)
Vertical: (3, 5/6), (-2, 5/13)
The shape of a Norman window can be approximated by a rectangle with a semicircle on top. What dimensions will admit the maximum amount of light if the perimeter of the window is fixed as P
w = 2P/(4 + pi)
h = P/(4+pi)
The position of a particle at time t is represented by s(t) = 1/3 * t^3 -3t^2 + 8t - 5. When is the particle at rest?
t = 4, t=2
Classify extrema using the first derivative test for f(x) = sin^2(x) + 1
Maxima: pi/2 + npi
Minima: npi
Find (f-1)`(b) for g(x) = (3x^8 + x^3 + 1)^(3/2) given b=g(1)
2/(8sqrt(5))
Find the coordinates when dy/dx = 2/3 for 3x^2 + 6 = 3xy
(sqrt(6), 4sqrt(6)/3)
(-sqrt(6), -4sqrt(6)/3)
A fisherman is reeling in a fish at a rate of 20 centimeters per second. If the tip of his fishing rod is 4.5 meters above water and we assume the fish lies along the top of the water the entire time, how fast is the fish approaching when 7.5 meters of fishing line are still out?
Approach at -.125 meters per second
Apply L'Hôpital's rule for 2x/(x2 -3x+4)
infinity = limit
Use the first and second derivative test to find increasing or decreasing intervals and points of inflection
Intervals: decreasing (-inf,-sqrt(3)/2), (sqrt(3)/2,inf)
increasing (-sqrt(3)/2,sqrt(3)/2)
inflection points: x = 0
Find (f-1)`(b) for g(x) = x^17 + 2x^11 -2x+3 for b = 4
1/37
Find a classify extrema for 3x^2 + 2y^2 = 16 by implicit differentiation
Max: (0, 2rt(2))
Min: (0, -2rt(2))
The position of a ball at time t is s(t) = t3-6t2+9t. Find when the acceleration when the velocity is equal to 0
a(1) = -6
a(3) = 6
What is the derivative of arccsc
-1/(|x|sqrt(x2-1))