Need help w/ a few Calc problew\ms.

[Stewdogg]

So this big list of problem are due in a week now and these 9 are stumping me, if anyone could help with all or any one of them I'd appreciate it. Here they are:

1. Sketch a graph of f(x) = x+4/x (that's a fraction) and label all intercepts and asymptotes. Then find and solve the difference quotion for f(x).

2. Find and simplify the difference quotients for the following:
(a) f(x)=x[squared]+14x-2
(b) g(x) [sqrt]4x-1 (4x-1 is inside the square)

3. Sketch a graph of the piecewise function: f(x)= X+4, x [less-than]-2
(x+2)[cubed], x[greater-than]-2

4. Simplify x[squared]+6x+5 / x[squared]+3x+2 (also a fraction)

5. Simplify e[squared]x(1)-2e[squared]x(x-3) / (e[squared]x)[squared] (also a fraction and and the x on the 2x's should be a superescript and in the powers)

6. Let f(x)=-3x-2. Find f-1(x) (inverse). What is the relationship between the slopes? Show the rises and the runs for the slowes of bothe f and f-1 on the same cooridinate axes.

7 The inverse function for f(x)=9x (that should be 9 tot he x power) is the logarithmic function f-1(inverse)(x)=log9x. Sketch graphs of both these functions on the same cooridinate axes. Then do the following:
(1) Connect the points (1,9) and (2, 81) on the graph of f(x)=9x with a line.
(2) Connect the points (9,1) and (81,2) on the graph of f-1(x)=log 9x with a line.
(3) Find the slopes of the two lines just drawn.
(4) Are the slopes in the two lanes related in thye same way as those in #6?

8. Solve the euation 5/x (fraction)x3/2 (x to the power 4/3)-2 (3/2 (fraction)x1/2 (x tot he 1/2)=0 for x.

9. An exponential function that closely models the number of millions of transistors in Pentium processors in the lates 1990s in the function T(x)=7.4932+0.0068e2.84462x (that's e to the power of that, including the x), where x is the number of years after 1997.

(a) Find the average rate of change for T(x) from x=0 to x=5.
(b) Find the average rate of change for T(x) from x=0 to x=2.
(c) Find the average rate of change for T(x) from x=0 to x=0.5.
(d) Find the average rate of change for T(x) from x=0 to x=0.1.

You have the slope of secant line sthat should approach the slope of the tangent line at x=0, which happens to have a value of approximately 0.019. Are your values above getting close and closer to 0.019?
(e) How accurate is this model in predicting the number of transistors in Pentium processors today (use x=9 for 2006)

 

[galactus]

2. Find and simplify the difference quotients for the following:
(a) \(\displaystyle f(x)=x^{2}+14x-2\)

I assume you know what the difference quotient is. Apply it. If it's correct, you should get the derivative of the quadratic. I'll step through this one and you do the others. Okey-doke?.

\(\displaystyle \L\\\lim_{h\to\0}\frac{f(x+h)-f(x)}{h}\)

\(\displaystyle \L\\\lim_{h\to\0}\frac{\underbrace{(x+h)^{2}+14(x+h)-2}_{\text{f(x+h)}}-\underbrace{(x^{2}+14x-2)}_{\text{f(x)}}}{h}\)

Simplify:

\(\displaystyle \L\\\lim_{h\to\0}\frac{x^{2}+2xh+h^{2}+14x+14h-2-x^{2}-14x+2}{h}\)

\(\displaystyle \L\\\lim_{h\to\0}\frac{2xh+h^{2}+14h}{h}\)

\(\displaystyle \L\\\lim_{h\to\0}2x+h+14\)

Now if h approaches 0, what is the limit. That's the derivative of your function. Is it?.