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  • MarkFL's Avatar
    Today, 01:11
    MarkFL replied to a thread Variation Expression in Pre-Calculus
    Let's let $F$ be the fixed costs and $M$ be the marginal cost (the cost to machine one part), and $x$ be the number of parts machined. Then the total...
    2 replies | 30 view(s)
  • MarkFL's Avatar
    Today, 00:39
    If you have a piece of wire whose length is $a$, and we bend it into a square, then each side of the square will be: \frac{a}{4} And so what...
    5 replies | 51 view(s)
  • MarkFL's Avatar
    Today, 00:15
    Let's go back to: f(x)=A(x-6)\left(x^2+1\right) Now, we set: f(2)=A(2-6)\left(2^2+1\right)=-20A=-10\implies A=\frac{1}{2} And so we have:
    5 replies | 53 view(s)
  • MarkFL's Avatar
    Yesterday, 23:27
    MarkFL replied to a thread Sum of the Roots in Pre-Calculus
    Here's another approach: Suppose we have: ax^2+bx+c=0 Them by the quadratic formula, we have that the sum $S$ of the roots is given by: ...
    3 replies | 31 view(s)
  • MarkFL's Avatar
    Yesterday, 23:22
    MarkFL replied to a thread Product of the Roots in Pre-Calculus
    Here's another approach: Suppose we have: ax^2+bx+c=0 Then by the quadratic formula, we know the product $P$ of the roots is: ...
    6 replies | 51 view(s)
  • MarkFL's Avatar
    March 27th, 2017, 12:24
    Just to follow up, here is the completed table: Sum $S$ Probability of $S$: $P(S)$ Net Gain/Loss (in dollars) $G$ Product $G\cdot P(S)$
    2 replies | 81 view(s)
  • MarkFL's Avatar
    March 26th, 2017, 22:32
    MarkFL replied to a thread Radius of Circle in Pre-Calculus
    That is the radius of the circle after it has been increased by $a$ units. If I tell you that my weight increased by 20 lbs., then you know my...
    4 replies | 84 view(s)
  • Jameson's Avatar
    March 26th, 2017, 22:19
    Hi tmt, (Wave) For the quoted part in bold, should this read "If all cars are reserved for the day..."? I think there are two events here,...
    1 replies | 49 view(s)
  • MarkFL's Avatar
    March 26th, 2017, 21:35
    MarkFL replied to a thread Radius of Circle in Pre-Calculus
    Let's let $0<a$ be the number units the radius must be increased. And so the change in area we can write as: \Delta A=\pi(r+a)^2-\pi r^2=b Now...
    4 replies | 84 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 21:53
    MarkFL replied to a thread The Distance Across in Geometry
    \overline{MK}=\sqrt{(\sqrt{2}a)^2+(\sqrt{2}b)^2}=\sqrt{2\left(a^2+b^2\right)}=\sqrt{2(50)}=\sqrt{100}=10 :D
    8 replies | 95 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 21:07
    MarkFL replied to a thread Lagrange Multipliers 2 in Calculus
    I arbitrarily chose another point on the constraint, so that we could do a comparison like I mentioned just now in the other thread. :D
    9 replies | 96 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 21:04
    MarkFL replied to a thread Lagrange Multipliers in Calculus
    I chose the point as it is on the constraint. Using that point, we can determine if our one critical point is a maximum or a minimum. If the...
    9 replies | 83 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 19:21
    MarkFL replied to a thread Lagrange Multipliers 2 in Calculus
    I agree that the point $(2,2)$, is the only one that meets all criteria. Now we need to compare the value of $f$ at another point on the constraint,...
    9 replies | 96 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 19:08
    MarkFL replied to a thread Lagrange Multipliers in Calculus
    I agree that of the 3 critical points, $(1,1)$ is the only one in quadrant I. Now, we know this is either a maximum or a minimum, and to determine...
    9 replies | 83 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 11:15
    MarkFL replied to a thread Factoring...6 in Pre-Calculus
    It might be more clear to state something like the following: The difference of cubes formula states: p^3-q^3=(p-q)\left(p^2+pq+q^2\right) ...
    5 replies | 79 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 10:47
    MarkFL replied to a thread Lagrange Multipliers 2 in Calculus
    Consider: e^u=0 What do you get when solving for $u$? Okay, you correctly found $x^2=y^2$...what do you get when you substitute for...
    9 replies | 96 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 10:39
    MarkFL replied to a thread Lagrange Multipliers in Calculus
    What I would do is use the constraint to determine $y=2-x$. Now substitute for $y$ in both equations you mentioned, and solve for $x$, then your...
    9 replies | 83 view(s)
  • MarkFL's Avatar
    March 25th, 2017, 02:30
    This is a calculus question...please don't continue to post calculus questions in other forums. If given: ...
    3 replies | 90 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 23:12
    MarkFL replied to a thread Lagrange Multipliers 2 in Calculus
    If you solve both equations for $\lambda$ and then equate the results, you obtain: \frac{ye^{xy}}{2x}=\frac{xe^{xy}}{2y} Multiply through by 2:...
    9 replies | 96 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 21:24
    MarkFL replied to a thread Lagrange Multipliers in Calculus
    Okay, so what this implies is: \frac{x}{\sqrt{6-x^2-y^2}}=\frac{y}{\sqrt{6-x^2-y^2}} Cross-multiply: x\sqrt{6-x^2-y^2}=y\sqrt{6-x^2-y^2} ...
    9 replies | 83 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 16:18
    MarkFL replied to a thread The Distance Across in Geometry
    Using the Pythagorean theorem, we find: \overline{MK}=\sqrt{(\sqrt{2}a)^2+(\sqrt{2}b)^2}=\sqrt{2\left(a^2+b^2\right)} Now, we know that...
    8 replies | 95 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 06:10
    MarkFL replied to a thread [SOLVED] Minimum of function under constraint in Calculus
    I would use W|A: W|A - optimize 2x+y subject to xy=18
    10 replies | 288 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 04:29
    The objective function is linear, so it describes a plane, and so I don't believe there will be any saddle points, or in fact any critical points...
    4 replies | 65 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 04:04
    MarkFL replied to a thread Boat direction in Calculus
    Is this a calculus-based physics course?
    6 replies | 102 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 01:37
    What I would do is observe that we have cyclical symmetry, that is we may interchange $x_1$ and $x_2$ with no change in either the objective function...
    4 replies | 65 view(s)
  • MarkFL's Avatar
    March 24th, 2017, 01:25
    MarkFL replied to a thread The Distance Across in Geometry
    Not quite...it would be \sqrt{2}a and \sqrt{2}b...so what would the diagonal of the rectangle be?
    8 replies | 95 view(s)
  • MarkFL's Avatar
    March 23rd, 2017, 21:17
    MarkFL replied to a thread The Distance Across in Geometry
    I would let $a$ be the length (in cm) of the segments with 1 tick mark, and $b$ be the length (in cm) of the segments with 2 tick marks. And so,...
    8 replies | 95 view(s)
  • MarkFL's Avatar
    March 23rd, 2017, 19:30
    MarkFL replied to a thread Boat direction in Calculus
    Being that it is an optimization problem, on a function in two variables, the way I worked it using the calculus is the most straightforward way I...
    6 replies | 102 view(s)
  • MarkFL's Avatar
    March 23rd, 2017, 12:53
    MarkFL replied to a thread Boat direction in Calculus
    I think what I would do is orient a coordinate system such that south is up and west is to the right. At time $t=0$, have boat A at the origin, and...
    6 replies | 102 view(s)
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