k for y = (4/3)x^3+2kx^2+5x+3; Trap. Rule w/ n = 4; etc

J

Jon.Monreal

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So I'm working on a set of problems to which there is no textbook, and I am having problems with a few of them. I have searched Wikipedia as well as the internet (for teacher's notes, etc.) to no avail for these problems, and am having problems setting them up, let alone solving them. I would very much appreciate any help.

1. "For what values of k does the graph of y=(4/3)x^3+2kx^2+5x+3 have two tangent lines parallel to the x-axis."

2. a)"Approximate the integral of x^2 dx where a=1 and b=4 using the trapezoidal rule with n=4."
b)"Find the maximum possible error in the answer to (a)."
c)"Find the numer of equal-width subdivisions required to approximate the integral with an error of less than 0.001."

3. "The base of a solid object is a circle with a radius of 4. Every verticle cross section is an equilateral triangle. What is the volume of the solid?"

4. "The base of a solid object is the region in the xy-plane bounded by the graphs of y=x^2 and y=4. Every verticle cross section parallel to the y-axis is a rectangle with a height of 3. What is the volume of the object?"

5. "The mean value theorem for integrals says that every continous function attains its average value on an interval at some point in the interval. For the function y=tanx, find some number x on the interval [(-pi/4),(pi/4)] such that f(c) equals the average calue of the function on the interval."

6. "Use differentials to estimate 4.1^4."
Click to expand...

Thanks.
 
1. This can be done with a first derivative. One must find the values of k such that there are TWO DISTINCT relative extrema.

dy/dx = 4x^2 + 4kx +5

This can be equated to zero and solve with the quadratic formula. It is clear from this, that two distinct solutions will be obtained for k^2 - 5 > 0
 
tkhunny said:
1. This can be done with a first derivative. One must find the values of k such that there are TWO DISTINCT relative extrema.

dy/dx = 4x^2 + 4kx +5

This can be equated to zero and solve with the quadratic formula. It is clear from this, that two distinct solutions will be obtained for k^2 - 5 > 0
Click to expand...

I can see now.

But is it -sqrt(5)>k>sqrt(5) or (-sqrt(5))/2>k>(sqrt(5))/2? This has me a bit confused.
 
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