April 24th, 2021

• physics calculus 08:21

  • The slope is the constant of proportionality of the line between two points. What does the slope or line function between two points represent? what would be the behaviour if we went along it.

  • GIven any function $y=f(x)$ we calculate the average rate of change of $y$ with respect to $x$ over the interval $[x_1,x_2]$ by dividing the change in the value of $y$ by the length of the interval over which the change occurs $h$

    • $\frac{f(x_1+h)-f(x_1)}{h}$

  • The limit value of a function does not depend on how the function is defined at the point being approached. The function is not always defined for a given $f(x)$ but we can still look at the limit

    • A limit may not exist if there is not a definite value being approached but could be two sided
    • Nice definition thats more rigourous defining the margin of error
    • The symbol “$x\to c^{+}$” means that we consider only values of x greater than c
    • At any x value if we can get a limit as x tends towards c we can say that x is continuous at c. There is no unexpected behaviour at c. Depending on if the limit is two sided etc. it can be right or left continuous also. The unpredictability of a value at a given point. Its a nice mixture of the idea of limits with continuity.
    • Intermediate Value Theorem If $f$ is a continuous function on a closed interval $[a,b]$ and if $y_0$ is any value between $f(a)$ and $f(b)$, then $y_0=f(c)$ for some c in $[a,b]$
      • “The proof of the Intermediate Value Theorem depends on the completeness property of the real number system”

  • The slope of a the curve at a point is the limit of a minor change in x and its output, if that limit exists.

  • The domain of a differentiable function is defined as the set of points $f$ for which the limit exists. If a point in differentiable for all in the domain of f we say $f$ is differentiable

  • differentiation is an operation performed on a function. The derivative of the function $f(x)$ with respect to the variable x is $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

    • Provided the limit exists
    • Another way to denote $f^{\prime }(x)$ is $\frac{d}{dx}f(x)$

  • Plotting the derivate allows us to analysis the slope for x’s in the initial graph.

  • Not having a derivative at a point, no finite limit. Or differing finite limits. differentiation is a smoothness condition on a graph. We can define a consistent rate of change. if f has a derivative at $x=c$ then f is continuous at $x=c$

  • constant functions never change and that the slope of a horizontal line is zero at every point

  • second derivative vs chain rule April 25th, 2021