MATH 1426 Jeopardy #1

Optimization

100

Identify the 6 steps to solve an Optimization Problem.

1.) Draw/Label a picture

2.) Create objective and constraint equations

3.) Substitute the constraint equation into the objective

4.) Find the first derivative and set it to zero

5.) Solve for variables

6.) Second Derivative Test to find the maxima/ minima

200

What are the first and second derivative test used for?

First derivative test= to find critical points

Second derivative test= to find if your answer is a max or a min

300

A rectangular field has an area of 600m^2. Fencing is required to enclose the field and to divide it into two equal halves in terms of the length x. Find the objective (F(x)) and constraint.

Constraint: xy=600

Objective: F(x)=3x+2y

400

!!!!FREE POINTS!!!!!!

EXTRA 400 POINTS

500

A square based box with a volume of 16 ft³, need to built with a minimum cost. Materials for the sides cost p dollars and the cost for the square bases (top and Bottom) cost 2p dollars. What would the dimensions of the crate be?

s= and h=4

Linear Approximation

100

Write the formula for Linear Approximation [L(x)=?]

L(x)=f(a)+f'(a)(x-a)

200

Find the L(x) for (2.01)^3 by finding a,x, and f(x).

8.12

300

Find L(x) for cos 92

-pi/9 or -0.0349

400

Find L(x) for sin 148

0.5302

500

Find L(x) for f(x)=x^(2/3) at a=8. Then use L(x) to estimate f(8.3).

4.1

Differentials

100

 What is the formula for differentials?

dy=f'(x)(x₂-x₁)

200

Let y=x^2+1. Find the differential dy if x changes from 6 to 5.9

-1.2

300

Radius of a circle is 4ft, w/ a possible error of +- 1/5 ft. Estimate the error in the calculates area.

+- 8pi/5

400

Let y= -2x^2+3. Find dy if x changes from 2 to 1.5.

4

Mean Value Theorem

100

What are the three conditions for a graph to be differentiable over an interval?

1.) The graph has to be continuous over an interval

2.) The graph can not have sharp corners 

3.) The graph can not have vertical lines or too many oscillations

200

If f is continuous on [a.b] and differentiable on (a,b), then there is at least a point c in (a,b), such that

f'(c)= f(b)-f(a)/b-a

300

Find f'(c) when f(x)=x³+x-1 on [0,2]

5

L'Hopital's Rule

100

What is L'Hopital's Rule? and when can it be used?

lim_x->a f'(x)/g'(x) only when 0/0 or (+/-infinity/infinity)

200

What is the indeterminate form of the limit

lim_x->(pi/2) (tan x)^{cos x}

infinity^0

300

lim_x->infinity (4x³-3)/(ln(x)+5)

positive infinity

400

lim_x->infinity x^{(5/x^2)}

0

500

lim_x->0 (sin x)^{tan x}

Hint: log diffrentiation and L'Hopitals' rule

1