Differential Calculus
"All The Mathematical Methods I Learned In My University Math Degree Became Obsolete In My Lifetime" - Keith Devlin
Formal
Definition of a Limit: Here we explore the delta-epsilon
definition of a limit, with E and D representing Epsilon and Delta
respectively, a the value that x approaches and L the proposed value of
the limit. For now f(x)=x, bit you can adjust this to any value
you like. See if you can verify that the limit L corresponds with
your intuition! (https://www.desmos.com/calculator/it3rxtvbkq)
Formal Def of Limit & Discontinuous Functions: At the points of discontinuity, explore the "obvious" candidates for a limit and see if any of them could be a limit. (https://www.desmos.com/calculator/hej5uitp3x)
Limit definition of derivative: Using the sliders to control the points defining the secant line, you can compare the slope of any secant line (m) with the derivative at the left most point. By changing f(x), you can see how the derivative of different functions behave. (Link https://www.desmos.com/calculator/xdkzaygq01)
Slopes and derivative: Here, you can see the tangent line at any given point, the slope of the tangent line, and how this changing slope defines a new function, the derivative! Use sliders to control the point of tangency and change f(x) to see this applied to a variety of functions. (Link https://www.desmos.com/calculator/hxqz6v1z4q)
Formal derivative: Of course, given a function f(x), we sometimes just want to know what it's derivative is (with respect to x):
Extreme Value Theorem: Given a continuous function defined on [a,b], the extrema (min & max) must occur, and for smooth functions they must occur at the end points or the critical points. To see a visualization of this, consider the graph below. f(x) can be adjusted to be any function, and there are sliders for a,b. See that the global extrema occur exactly as described above. (https://www.desmos.com/calculator/uvth2liu7t)
Mean Value Theorem: Enter your function and end points a and b, and you can see where within [a,b], there are points c where f'(c)=(f(b)-f(a))/(b-a). (https://www.desmos.com/calculator/rvnujzmweb)
Finding Critical Points: Sometimes, we just want to find critical points, no fuss, no bs. One can replace 0 with any other value to find points where the derivative of f(x) has those values:
Newton's Method: This calculator doesn't automatically do Newtons Method, but f(x) (i.e. the red curve) is your function, and the blue line is the tangent line generated by your current value for x_n (represented by a). Click on the blue like to see where the x-intercept occurs and update your x_n (a). https://www.desmos.com/calculator/lfm51is0ba
Newton's Method automated: Since Newtons method is defined as a recursive loop, it makes sense to automate the process. Enter the function whose roots you wish to find as f(x), and your intital guess as initialx:
Formal Def of Limit & Discontinuous Functions: At the points of discontinuity, explore the "obvious" candidates for a limit and see if any of them could be a limit. (https://www.desmos.com/calculator/hej5uitp3x)
Limit definition of derivative: Using the sliders to control the points defining the secant line, you can compare the slope of any secant line (m) with the derivative at the left most point. By changing f(x), you can see how the derivative of different functions behave. (Link https://www.desmos.com/calculator/xdkzaygq01)
Slopes and derivative: Here, you can see the tangent line at any given point, the slope of the tangent line, and how this changing slope defines a new function, the derivative! Use sliders to control the point of tangency and change f(x) to see this applied to a variety of functions. (Link https://www.desmos.com/calculator/hxqz6v1z4q)
Formal derivative: Of course, given a function f(x), we sometimes just want to know what it's derivative is (with respect to x):
Extreme Value Theorem: Given a continuous function defined on [a,b], the extrema (min & max) must occur, and for smooth functions they must occur at the end points or the critical points. To see a visualization of this, consider the graph below. f(x) can be adjusted to be any function, and there are sliders for a,b. See that the global extrema occur exactly as described above. (https://www.desmos.com/calculator/uvth2liu7t)
Mean Value Theorem: Enter your function and end points a and b, and you can see where within [a,b], there are points c where f'(c)=(f(b)-f(a))/(b-a). (https://www.desmos.com/calculator/rvnujzmweb)
Finding Critical Points: Sometimes, we just want to find critical points, no fuss, no bs. One can replace 0 with any other value to find points where the derivative of f(x) has those values:
Newton's Method: This calculator doesn't automatically do Newtons Method, but f(x) (i.e. the red curve) is your function, and the blue line is the tangent line generated by your current value for x_n (represented by a). Click on the blue like to see where the x-intercept occurs and update your x_n (a). https://www.desmos.com/calculator/lfm51is0ba
Newton's Method automated: Since Newtons method is defined as a recursive loop, it makes sense to automate the process. Enter the function whose roots you wish to find as f(x), and your intital guess as initialx: