Linearized Station
Find a linearization of
f(x) = 3xe2x-10 @ x=5
What is
L(x) = 33x - 150
f(x) = 6x5 + 33x4 - 30x3 + 100
What are
x = -5, 0, 3/5
Determine all intervals where the following function is increasing or decreasing
f(x0 = -x5 + 5/2 x4 + 40/3 x3 + 5
What is
increase: -2 < x < 0 and 0 < x < 4
decrease: -∞ < x < -2 and 4 < x < ∞
We are going to fence in a rectangular field. If we look at the field from above the cost of the vertical sides are $10/ft, the cost of the bottom is $2/ft and the cost of the top is $7/ft. If we have $700 determine the dimensions of the field that will maximize the enclosed area.
What is
x = 350/9 , y = 35/2
limx -> 2 (x3 - 7x2 + 10x) / (x2 + x - 6)
What is -6/5
Find the linearization of
f(x) = [√ (2x)] [ln√ (x-1)]
centered at x = 2, then use the linearization to estimate f(2.1)
What is
L(x) = 2x - 4
f(2.1) = 0.2
Determine the absolute extrema for the following function and interval
g(t) = 2t3 + 3t2 - 12t + 4 on [-4,2]
What is
g(t) = 24 @ t = -2 ;
g(t) = -28 @ t = -4
Determine the intervals on which the function increases and decreases
f(x) = 2x3 - 9x2 - 60x
What is
increasing: (−∞,−2) & (5,∞)
decreasing: (−2,5)
We want to build a box whose base length is 6 times the base width and the box will enclose 20 in3. The cost of the material of the sides is $3/in2 and the cost of the top and bottom is $15/in2. Determine the dimensions of the box that will minimize the cost.
V=lwh
What is
w = 0.7299
l = 4.3794
h = 6.2568
Evaluate limx-> -4 sin(πx) / (x2 - 16)
What is -π /8
What is the linearization of
f(x) = x1/3 @ x = 8
What is
L(x) = (1/12) x + 4/3
Determine all the critical points for the function
f(x) = x2 ln(3x) + 6
What is
x = 1/ (3√ e)
h(t) = 50 + 40t3 - 5t4 - 4t5
What is
increasing: (−3,0) & (0,2)
decreasing: (−∞,−3) & (2,∞)
We have 45 m2 of material to build a box with a square base and no top. Determine the dimensions of the box that will maximize the enclosed volume.
V = lwh
What is
l = w = 3.8730
h = 1.9365
Evaluate limx-> -∞ x2 / e1-x
What is 0
What is the linearization of
g(z) = z1/4 @ z = 2
What is
L(z) = 21/4 + (1/4) (2-3/4) (z - 2)
Determine the absolute extrema of
R(x) = ln(x2 + 4x + 14) on [-4,2]
(USE CALCULATOR)
What is
absolute max: 3.2581 @ x=2
absolute min: 2.3026 @ x = -2
Given that f(x) and g(x) are increasing functions. If we define h(x)=f(x)+g(x) show that h(x) is an increasing function
h'(x) = f'(x) + g'(x)
f'(x) > 0 and g'(x) > 0
then h'(x) > 0
A manufacturer needs to make a cylindrical can that will hold 1.5 liters of liquid. Determine the dimensions of the can that will minimize the amount of material used in its construction.
V = πhr2 ; A = 2πrh
r = 6.2035
h = 12.4070
Evaluate limx-> 0+ [x2 ln(4x2)]
What is 0
Find the linearization of
h(t) = t4 - 6t3 + 3t - 7 @ t=-3
(USE CALCULATOR)
What is
L(t) = -267t - 574
Suppose that the amount of money in a bank account after t years is given by:
A(t) = 2000 - 10t e5 - (t^2)/8
Determine the minimum and maximum amount of money in the account during first 10 years. (USE CALCULATOR)
What is
max will be $2000 @ t=0 ;
min will be $199.66 @ 2 years
Evaluate limx->∞ [ex + x]1/x
What is e