April 25th, 2021

• physics calculus 07:15
  • A function might not have an absolute min/max if the function is unbounded at one of the endpoints
  • A function must also be finite
  • An interior point on the domain of a function whose derivative is zero is a **critical point **(inflection, max, min, local/absolute)
  • Rolle’s Theorem if there is a horizontal line touching the graph twice then there is a point in this interval whose derivative is zero
  • Can be used the prove the mean value theorem Suppose $y=f(x)$ is continuous over a closed interval $[a,b]$ and differentiable on the interval’s interior $(a,b)$. Then there is at least one point $c$ in $(a,b)$ at which $\frac{f(b)-f(a)}{b-a}=f^{\prime }(c)$
    • The slope of the line between a and b will be repeated or exist as the derivate at a point c in the interval
    • At some point along the average change between two points the instantaneous change must equal the average change over that interval.
    • Corollary from mean value theorem If $f^{\prime }(x)=g^{\prime }(x)$ at each point $x$ in an open interval $(a,b)$, then there exists a constant $C$ such that $f(x)=g(x)+C$ for all $x\in (a,b)$ that is, $f-g$ is a constant function on $(a,b)$
    • The graphs of functions with identical derivatives on an interval can only differ by a vertical shift. The constant is added to $f(x)$ not to the domain (horizontal shift)
    • The interval does not have to be finite
  • antiderivative function on an interval $I$ where $F^{\prime }(x)=f(x)$ for all $x$ in $I$ if $f(x)$ was the derivative of a function, what would that function be
    • If $F$ is an antiderivative of f on an interval I, then the most general antiderivative of $f$ on $I$ is $F(X)+C$
    • The function $F(x)=x^2$ is not the only function whose derivative is $2x$. The function $x^2+1$ has the same derivative. Any two antiderivatives differ by a constant. The functions $x^2+C$ where $C$ is an arbitrary constant form all the antiderivatives of $f(x)=2x$ If the derivate is the same they should just differ by a constant
    • How do we get this corollary from the mean value theorem? ?
    • Finding the family of functions gives the general solution to a problem of a differential equation $\frac{dy}{dx}=f(x)$
      • Then we can solve for an initial value where $y(x_0)=y_0$ for some initial condition $x$
    • $\int f(x)dx$ the collection of antiderivates of f, or the indefinite integral of $f$ wrt $x$
  • definite integral
    • We say that a number $J$ is the definite integral of $f$ over $[a,b]$ and that $J$ is the limit of the riemann sums $J={\lim }_{|P|\to 0}{\sum }_{k=1}^{n}f(c_k)\Delta x_k$
    • $J\to {\int }_a^{b}f(x)dx$
    • Leibniz envisioned an infitied sum of function values multiplied by infinitesimal subintervals of dx. So $f(x)$ represents a continuous selection of function values (as opposed to $f(c_k)$). The symbol then for $J$ (the definite integral that is approached by the riemann sum) is ${\int }_a^{b}f(x)dx$
    • mean value theorem for definite integrals if $f$ is continuous on $[a,b]$, then at some point c in $[a,b]$, $f(c)=\frac{1}{b-a}{\int }_a^{b}f(x)dx\le maxf$
  • fundamental theorem of calculus
    • Part 1
      • $F^{\prime }(x)=\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$
      • Proved p.331
      • Saying the antiderivative is equal to the definite integral
    • Part 2
      • Eliminating the need to calculate the riemann sum
      • ${\int }_a^{b}f(x)dx=F(b)-F(a)$ If $f$ is continuous over $[a,b]$ and $F$ is any antiderivative of $f$ over $[a,b]$
      • Proved on p.332. Mad how implication can give us these insights, that to me at least, dont make intuitive sense.
    • Also have the net change theorem
• what is e? Still dont fully understand but know that its an exponential number that is proportional to its own derivative. Its a useful way to represent exponential growth