Cal Poly Pomona The Price Elasticity of Demand Questions

application of the derivative, Finding critical numbers, Extrema, Concavity, Curve sketching, Implicit Differentiation.

6. Find all errors in the following calculation. (state the line and the mistake)
Solving for the third derivative of the function f(x)
8. For the following demand function, find E and values of p (if any) at which total revenue
is maximized: a 500 10p.
–
=
-4x
1+ 2x
Line 1: f(x)
9. Find all errors in the following calculation. (state the line and the mistake)
Using implicit differentiation to find dy/dx given the function: (xy)4 = (y + y)?.
=
Line 2: f ‘(x)
=
2
3
=
Go
Line 3: f ‘(x)
=
2
Line 4: f ‘(x)
=
2
5
=
Line 5: f ‘(x)
=
Line 6: f ‘(x)
=
-4x
1 + 2x
-4(1 + 2x) – (2) – 4x
(1 + 2x)
-4 – 8x – 2 – 4x
(1 + 2x)
-6 – 12x
(1 + 2x)?
-6(1 + 2x)
(1 + 2x)?
-6
1 + 2x
(2)(-6) – 0(1 + 2x)
(1 + 2x)?
-12
(1 + 2x)
0(1 + 2x)? – 2(1 + 2x)(-12)
((1 + 2x)32
0 – 24(1 + 2x)
((1 + 2x)32
– 24(1 + 2x)
(1 + 2x)*
– 24
(1 + 2x)
dy
=
dx
dx
Line 7: F”(x)
=
4 5
5 4
Line 1: (xy)4 + y)
Line 2: [(xy)*] = [ (vº + y)?]
Line 3: 4(xy)*(1.y + x 4) = [(y° + y)^
Line 4 4(xy)*(1 · y + x
4*1 ) 3(y + y){(6y
any
+ 1)
Line 5: 4x*y*(y + x dy) = 360º + y)²(6y”. m + 1)
Line 6: 4x*y + 4x”yt
3(yº + y)²(6y5. dy
+ 1)
Line 7: 4x*y + 4xºy
(jº +
y)(18y” y + 3)
Line 8: 4x y + 4x y
18y” v + y). Saya + 3(yº + y)?
Line 9: 4x y + 4x y
18yº (v + y)? day 3(v + y)?
Line 10: 4x´y
18yº (yº + y)? . Play 4x*y; – 3(yº + y)?
Line 11: dy [4x®y4 – 18y°(+ y)*1 = 4x*y” – 3(y + y)?
4x+y^ – 3(y^ + y)
Line 12:
4xº
ty* – 18yºyº + y)^
=
Line 8: f “(x)
dy
dx
dy
dx
=
2
4 5
5 4
=
4 5
5 4
Line 9: f ‘(x)
=
dy
dx
=
5 4
dy
dx
Line 10: f'(x)
=
=
Line 11: f'(x)
=
4
2
dy
dx
=
Line 12: f ”(x)
=
3
7. I am making a rectangular pen with two additional fences across its width to store my
unruly nieces and nephews (see figure below). Find the maximum area possible if I
only have 36 m of fencing available. (Area = bh, Perimeter = 2b + 4h)
10. When Goku fought Frieza, he had no choice but to use the Spirit Bomb. A Spirit
Bomb is a blue energy sphere that absorbs the energy of anything living and willing to
give up the energy. As it absorbs more energy, the radius of it changes at a rate of 6 ft
per second. Find the rate of change of the volume of the Spirit Bomb when the radius
is 10 ft. (make sure to write the proper units)
V = ntr
2. HBO has released their revenue function R(x) where x represents the amount of
money spent on sub-par writers. This amount can range from $0 to $100. Find the
maximum revenue and the amount of money that should be spent on sub-par writers to
maximize their revenue when:
1600
R(x)
= 200
х
X + 8
3. Suppose G(t) is the price of GME stock at time t during a particular day.
a. The stock price begins to drop faster and faster. Is G'(t) positive or negative?
Why?
b. As the stock continues to drop faster, is G”(t) positive or negative? Why?
4. Sketch the graph of a function that has all of the following properties:
(a) Continuous for all real numbers
(b) f'(x) < 0 on (-00,-4) and (1,5) (c) f'(x) > 0 on (-4,1) and (5,00)
(d) f”(x) > 0) on (-00, – 4) and (5,00)
(e) F”(x) < 0 on (-4,5) (f) A y-intercept at (0, 2) 5. The Lucky Smells Lumber Mill sells 180,000 logs annually. It costs $16 to store each log for 1 year and $25 to place a reorder. Find the number of logs that should be produced per order, and how often they should place orders.