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pageSymbol Pushers Page
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| Back to Start Page |
| If you think you're good at symbol
pushing, try these... Prob1 - Prob2 - Prob3 - Prob4 - Prob5 5 is a slightly different version of Prob3 Also, try the ones below Common Mistake Review and Common Mistakes Plus2 Calculus Review that is prep for Differential Equations (post Calc II level) |
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| Problem
1. Solve for y. Recall that you
can't have a y in the answer! Note: Only intermediate algebra skills are required. Also, do this one without the aid of formulas or a calculator. |
| x2 + 4xy + 4y2 - 5x - y = 3 |
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Answer
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| click for Solution to 1 if you can't get the answer: top |
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| There are, of course, other ways to do these. |
| If you did these without the aid
of a calculator or formulas, you are a good symbol pusher in my opinion. |
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Problem
2.
Find the inverse of f (x)
hyperbolic sine
Note: Only college algebra skills are required. |
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Answer
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| click for Solution to 2 if you can't get the answer: top |
| . |
| There are, of course, other ways to do these. |
| If you did these without the aid
of a calculator or formulas, you are a good symbol pusher in my opinion. |
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Problem
3.
Answer
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| click for Solution to 3 if you can't get the answer: top |
| . |
| There are, of course, other ways to do these. |
| If you did these without the aid
of a calculator or formulas, you are a good symbol pusher in my opinion. |
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Problem
4 Answer
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| click for Solution to 4 if you can't get the answer: top |
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Problem
5 Answer
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| click for Solution to 5 if you can't get the answer: top |
| . |
| There are, of course, other ways to do these. |
| If you did these without the aid
of a calculator or formulas, you are a good symbol pusher in my opinion. |
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Solution to problem 1: Go to Top
| x2 + 4xy + 4y2 - 5x - y = 3 | problem |
| y2 + xy + x2/4 - y/4 = 5x/4 + 3/4 | divide by 4 and move terms |
| (y + x/2)2 - y/4 = 5x/4 + 3/4 | factor |
| (y + x/2)2 - (y + x/2)/4 = -x/8 + 5x/4 + 3/4 | add -x/8 to both sides to obtain form (z)2 - (z)/4 where z = y + x/2 |
| (y + x/2)2 - (y + x/2)/4 + 1/64 = -x/8 + 5x/4 + 3/4 + 1/64 | complete the square on z = y + x/4 square completer is [(1/2)(1/4)]2 = 1/64 |
| [(y + x/2) - 1/8]2 = 9x/8 + 49/64 | write in square form and simplify right side |
| (y + x/2) - 1/8 = ± (9x/8 + 49/64)1/2 | take the square root of both sides (using 1/2 in place of radical) |
| y = - x/2 + 1/8 ± (9x/8 + 49/64)1/2 | solve for y |
| y = - 4x/8 + 1/8 ± (72x/64 + 49/64)1/2 | prepare to put over common denominator 8 |
| y = - 4x/8 + 1/8 ± (72x + 49)1/2/8 | put over common denominators 8 |
| y = [- 4x + 1 ± (72x + 49)1/2]/8 | put all over common denominator 8 |
| y = - 4x
+ 1 ± (72x + 49)1/2 8 |
re-write |
Go to Top
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Solution to problem 2: Go to Top
| Find the inverse of f (x) | |
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original problem |
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let y = f (x) |
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interchange x and y |
Let y = ln u note: u = e y
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make a substitution |
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ln and exponential base e are inverses |
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multiply by 2 multiply by u get everything on left side to solve for u |
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complete the square in u |
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get ready to take square root |
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take square root |
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re-write |
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re-substitute |
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ln and exponential base e are inverses |
Throw out
(negative log) |
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Thus for all x. |
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.Solution to problem 3: Go to Top
Find the exact
solution of
Remove any imbedded radicals from the solution (no radicals within
radicals).
Solution:
Go to Top
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Solution to problem 4: Go to Top
Integrate
Let c = - c12 (for reasons apparent later).
Then,
replacing c, we get
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Let
An
alternate solution is when c = - c12 is not
used is
Go to Top
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Solution
to problem 5: Go to Top
Solve
removing
any radicals within radicals.
Use
the quadratic formula and complete the square.
Go to Top
