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Doug C.'s Resources

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Hi Don,   We know that alnx can be written as lnxa. That is the property to use.   So 2ln |x| can be written as ln |x|2, but now the absolute value symbol is not required because even if x is negative, after it is squared we will be taking the natural log of a postive number...

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Hi Julia,   If log2(3) is intended ti convey log of 3 base 2 (as in log23), then using the change of base formula is what will help you simplify/evaluate this expression.   log23 = log 3/log 2, log34 = log 4/log 3 and so on. If you write out several terms you will see a...

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Hi Don,   This feels like the area between two curves. Check out this graph then see if you can figure out what to do by finding antiderivative, etc.   https://www.desmos.com/calculator/wb3bmfhwuc    

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Did you know that for example the sum from 1 to 6 can be found by taking n(n+1)/6 where n = 6. Try it.    So one way to find the sum of 7 to 508 is to calculate 508(509)/2, then subtract the sum from 1 to 6.

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Here is another way to look at it--transform left side into right side.   sinx + sin (3x)   sinx + sin (2x +x)   sinx + sin2xcosx + cos2x sinx   (use sin(A+B) identity)   sinx + 2sinxcosxcosx + (2cos2x-1)sinx...

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The points in the plane that are 10 units from the point (2,-3) lie on a circle with center at (2,-3) with a radius of 10.   The equation for the circle is (x-2)2 + (y+3)2 = 100. So you want to find the points of intersection of that circle with the line y = 3. Use substitution,...

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Great question. Here is an article that discusses in detail.   https://www.theproblemsite.com/ask/2016/12/vinculum-and-bodmas   The key is that the line separating the numerator from the denominator of a fraction (called the vinculum) is itself a grouping symbol.    So...

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Hi Regina, I will get you started on question 2. No diagram for question. although my guess is you are folding that corner to touch the bottom of the page somewhere along its edge.   2. The length of the 3rd side depends on the size of the included angle. The formula for that relationship...

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Hi John,   In order to do this problem using the limit of a difference quotient you need to be aware of two theorems on limits involving sin and cos.    1. limt->0 (sin t)/t  = 1   2. limt->0 (1 - cos t)/t = 0   Assuming you have...

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Hi Prashant,    I am making some assumptions here: 1) you make no mention of calculus so I assume you do not have that at your disposal to determine when a function has a maximum. 2) Perhaps you know that for a rectangle satisfying the given conditions to generate a maximum area...

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Hi Don,   In order to discuss this easily let f(x) = (2x+5)3 and g(x) = (6x-1)5.   So we are going to use the product rule fg' + gf'.   fg' = (2x+5)3 5(6x-1)4 (6)  (note the use of the chain rule to get that factor of 6) -- also, (5)(6) = 30 gf'...

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Hi Don,   Need to find the slope of the tangent line at (2/3, 4/3). Since the original equation cannot be easily solved for y, use implicit differentiation. The understanding is that y is some function of x, whenever we take the derivative of y it is with respect to x, i.e. dy/dx. In...

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Geometry Problem (answer)

Hi Juan, Here are some suggestions. Try to determine the equations of the perpendicular bisectors of two sides of the triangle. That will require finding the midpoint of two of the sides and the slope of two of the sides. The perpendicular bisectors will have slopes that are negative reciprocals...

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