Tutor profile: Israel H.
1. What is the slope of the equation y = x? 2. What is the tangent of 45 deg? 3. What is the derivative of the equation y = x?
The answer to all 3 questions is 1. Is this a coincidence? Nope. Graph y = x on a Cartesian coordinate system. What is the angle of this line with respect to the x-axis? 45 degrees, right? And the tangent of an angle is "opposite over adjacent", which is the same as the slope of the line--rise over run. "Opposite" is rise, or change in y. And "adjacent" is run, or change in x. The derivative is a gradient function, and for question 3, this derivative is simply 1, the same as the slope of that line. In this discussion, the answer is the same for all 3 questions because all 3 questions are talking about the same thing.
The volume of a cylinder is given by the formula V = π r^2h. Find the maximum value of V if r + h = 12.
The condition says the height plus the radius is 12. This helps us simplify the volume formula from two variables to one variable Let's solve h+r = 12 in terms of r so that we can substitute h into the volume formula. Given h + r = 12, h = 12 - r when we subtract r on both sides. Now we can substitute for h into the volume formula. V = π r^2(12 - r). Expand this expression using the distributive property of multiplication. V = 12 π r^2 - π r^3 Since we're looking for a maximum, we need to find the derivative of V with respect to r. dV/dr = 24π r - 3π r^2. Set this equal to zero to find the maximum volume. dV/dr = 24π r - 3π r^2 = 0; Solve for r. First, factor an r: r(24π - 3π r) = 0. r = 0 is one solution to the above expression but not the answer we're looking for. r must be >0. So, 24π - 3π r = 0. Add 3πr to both sides. 24π = 3πr Divide both sides by 3π. r = 8. Now, to answer the question, what is the maximum volume? It is when r = 8. Plug that into the volume formula. (By the way, h is 4, since h = 12 - r = 12 -8.) V = π r^2 * h V = π (8)^2 * 4 = 256π. Graph V = (12 π r^2 - π r^3) so that you can see that V hits a maximum (for r >0) at r = 8.
I don't understand algebra. Why does it use symbols/letters?
I'm sure you've seen x and y used as "variables" in algebra, and while this may seem complicated at first, these symbols are part of the language of math. They make speaking the language much easier. But, like any language, you must first learn the basics and build on that in order to speak that language. Take for example this problem that I remember from 3rd grade. 5 + __ = 12 I'm sure you've seen a problem like this from back in grade school. The answer, of course, is 7. And guess what? This is algebra. It's just that, as we've matured, and now we're all grown up, we are ready for variables instead of filling in the blanks. Like so: 5 + x = 12 What is x? Just as the blank line above, x is 7. Or, in math language, x = 7.
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