Cal 1 Test 4 (3.11-4.6)

Linearized Station

100

Find a linearization of 

f(x) = 3xe^2x-10 @ x=5

What is 

L(x) = 33x - 150

200

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

300

What is the linearization of 

f(x) = x^1/3 @ x = 8

What is 

L(x) = (1/12) x + 4/3

400

What is the linearization of 

g(z) = z^1/4 @ z = 2

What is 

L(z) = 21/4 + (1/4) (2-3/4) (z - 2)

500

Find the linearization of 

h(t) = t⁴ - 6t³ + 3t - 7 @ t=-3

(USE CALCULATOR)

What is 

L(t) = -267t - 574

Critical Extremes

100

Determine all the critical points for the function:

f(x) = 6x⁵ + 33x⁴ - 30x³ + 100

What are 

x = -5, 0, 3/5

200

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

300

Determine all the critical points for the function

f(x) = x² ln(3x) + 6

What is 

x = 1/ (3√ e)

400

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

500

Suppose that the amount of money in a bank account after t years is given by: 

A(t) = 2000 - 10t e^{5 - (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

Cons of Cavities

100

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 < ∞ 

200

Determine the intervals on which the function increases and decreases 

f(x) = 2x³ - 9x² - 60x

What is 

increasing: (−∞,−2) & (5,∞)

decreasing: (−2,5) 

300

Determine the intervals on which the function increases and decreases 

h(t) = 50 + 40t³ - 5t⁴ - 4t⁵

What is 

increasing: (−3,0) & (0,2)

decreasing: (−∞,−3) & (2,∞)

400

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

What is 

h'(x) = f'(x) + g'(x)

f'(x) > 0 and g'(x) > 0

then h'(x) > 0

Optimized Prime (calculator)

100

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

200

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

300

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

400

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 

What is 

r = 6.2035

h = 12.4070

Going to L'Hospital

100

lim_{x -> 2} (x³ - 7x² + 10x) / (x² + x - 6)

What is -6/5

200

Evaluate lim_{x-> -4} sin(πx) / (x² - 16)

What is -π /8

300

Evaluate lim_{x-> -∞} x² / e^1-x

What is 0

400

Evaluate lim_{x-> 0+} [x² ln(4x²)]

What is 0

500

Evaluate lim_x->∞ [e^x + x]^1/x

What is e