1. ## Simple Calculus

My math class has just started getting into calculus and I have a few questions that I'd like to ask. Our school uses a program called TLM that spits out some random math problems from a huge database, and here are a couple of questions that I'm having trouble with.

Given the function $\displaystyle f(x) = 4x^2-1x+4$
evaluate $\displaystyle f(2+\Delta x)$

For this question should I just pop in $\displaystyle 2+\Delta x$ for x?

EDIT:

$\displaystyle y=5x^2+6x+21$

The question then asks me what is $\displaystyle \Delta Y$ and $\displaystyle \Delta y / \Delta x$ when x changes from 1.6 to 2.2
For delta y I just subbed in the numbers and got 15

For finding the dy/dx I'm drawing a complete blank. I feel like it should be easy but I can't figure out what to do.

2. You are correct for the first question:

$\displaystyle f(2+\Delta x)=4(2+\Delta x)^2-(2+\Delta x)+4.$

Your answer to the second question also looks correct. To find $\displaystyle \frac{\Delta y}{\Delta x}$, we just divide the answer by $\displaystyle \Delta x=2.2-1.6$.

To find $\displaystyle \frac{dy}{dx}$, we evaluate what is called a limit:

$\displaystyle \frac{dy}{dx}=\lim_{\small \Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}.$

We can do this by simplifying the fraction and noting that certain terms containing $\displaystyle \Delta x$ approach $\displaystyle 0$ as $\displaystyle \Delta x$ itself approaches zero, leaving us with the derivative of the function $\displaystyle f(x)$.

Hope this helps!

3. Thanks for the answers Scott, but I'm still having a bit of trouble grasping the concept for the second part of my second question.

$\displaystyle y=5x^2+6x+21$ when x changes from 1.6 to 2.2.

Find $\displaystyle \frac {\Delta y} {\Delta x}$

I don't really understand what "rule" or equation to use to find the answer. It ended up being 25.

Again, any help is much appreciated!

EDIT: I found out this is the delta process, but I'm still attempting to wrap my head around it.