Intro To Calculus
Table of Contents
- Derivatives
- Slope Segments
- Derivatives (Intro)-2
- Intuitive Derivative Graphing Exercise
- Intuitive Derivative Graphing Exercise: REVAMPED!!!
- Parabola As Line Segments
- Tangent To A Parabola
- Derivative With Secant Lines
- Sketch Derivative Graphs
- Riemann Sums
- Limits For Derivatives
- Derivative With Limit
- Optimization
- Optimization of the volume of a box
- Optimization Cylinder Maximize Volume
- Kopija od Optimization - Box
- Optimize Volume Of Square Prism Given Surface Area
- Optimize Surface Area Square Prism Given Volume
- Second Derivative
- First Derivative and Second Derivative of a Function
- Which is f, f', f''? Part 2 !!!
- Which is f, f', f'' ?
- Second Derivative Plot
- Distance / Velocity / Acceleration Graphs
- Distance Velocity Acceleration Piecewise
- More Complicated Derivatives
- Derivative of Sine & Cosine Functions (Quick Investigation)
- When does f'(x) = f(x)?
- Product rule change in area
- Chain Rule dy/dt
- Related Rates
- Falling Ladder !!!
- Related Rates (1)
- Related Rates (2)
- Related Rates - Square
- Related Rates - Circle 1
- Related Rates - Circle 2
- Related Rates: Adjustable Cone with dh/dt Constant
- Cone - Related Rates
- Related Rates: Filling/Draining Cone
- Integrals
- Riemann Sums
- Definite Integral Illustrator (I)
- Riemann Sums
- Riemann Sums
- Area Between Curves
- Applications Of Integrals
- Finding areas between curves
- Areas Between Curves
- Area Between Two Curves Visualization
- Volume of Solid of Revolution about x-axis
- Disk/washer method for finding volume of solid of revolution
- Volumes of Solids - Disk & Washer Method
- Solid of revolution about y=b axis
- Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions
- Calculus - Arc Length
- Arc Length Visualization
Slope Segments
Optimization of the volume of a box
update of a 2013 ggb file
First Derivative and Second Derivative of a Function
This applet displays a function f(x), its derivative f '(x) and its second derivative f ''(x). Grab open blue circles to modify the function f(x). [br]Grab a solid circle to move a "test point" along the f(x) graph or along the f '(x) graph.
Move the open circles around to adjust for the curve you want. Try selecting boxes, moving points and so on until you become comfortable with the mechanics of this applet. Then select some boxes as needed in order to focus on particular comparisons and relationships.
Derivative of Sine & Cosine Functions (Quick Investigation)
In the applets below, graphs of the functions [math]f\left(x\right)=sin\left(x\right)[/math] and [math]f\left(x\right)=cos\left(x\right)[/math] are shown. [br]In each applet, drag the BIG WHITE POINT along the graph of the displayed function. [br][br]The y-coordinate of the point being traced out = the slope of the tangent line to the graph of f. [br]Interact with each applet for a few minutes, then answer the questions that follow.
1)
Based on your observations, if [math]f\left(x\right)=sin\left(x\right)[/math], can you write an expression for [math]f'\left(x\right)[/math]?
2)
Based on your observations, if [math]f\left(x\right)=cos\left(x\right)[/math], can you write an expression for [math]f'\left(x\right)[/math]?
3)
Use the limit-definition of a derivative to prove that if [math]f\left(x\right)=sin\left(x\right)[/math], then [math]f'\left(x\right)=cos\left(x\right)[/math].
4)
Use the limit-definition of a derivative to prove that if [math]f\left(x\right)=cos\left(x\right)[/math], then [math]f'\left(x\right)=-sin\left(x\right)[/math].
Falling Ladder !!!
Suppose a ladder that's 10 feet long is (somehow) resting up vertically against a wall. [br][br]The bottom of the ladder is then kicked out so that the base of the ladder is moving away from the wall at a rate of 3 ft/sec. (Go ahead and [color=#38761d][b]kick the ladder[/b][/color]). [br][br][color=#9900ff][b]At what rate is the ladder's height, [i]h[/i], changing[/b][/color] when the bottom of the ladder is 6 feet away from the wall? 9 feet away from the wall? [br][br]Use implicit differentiation to determine the answers to these 2 questions, and then check the approximate values of your your results within the applet. [br][br]In fact, at any time, you can adjust the values of [i]x[/i] and [math]\frac{dx}{dt}[/math]. [br][br]
1.
[color=#9900ff][b]Why is the value of [/b][/color][math]\frac{dh}{dt}[/math][color=#9900ff][b] always negative[/b][/color] (except at [i]x[/i] = 0)? Explain.
2.
How far away does the base of the ladder need to be away from the wall in order for [math]\frac{dx}{dt}=\left|\frac{dh}{dt}\right|[/math]? You can guess-and-check using the applet above. Yet be sure to use calculus to obtain an exact solution!
Quick (Silent) Demo
Finding areas between curves
Explore the area under the line and the area under the quadratic. Then look at both at the same time to see how to find the area between two curves.
Will this technique work to find the area between any two curves?[br]What if one of the curves is below the x-axis, will this technique still work?[br]How will you find the limits of integration for the two functions?
Summary of area between two curves
We know that the area is the quantity which is used to express the [br]region occupied by the two-dimensional shapes in the planar lamina. In [br]calculus, the evaluate the area between two curves, it is necessary to [br]determine the difference of definite integrals of a function. The area [br]between the two curves or function is defined as the [url=https://byjus.com/maths/definite-integral/]definite integra[/url]l[br] of one function (say f(x)) minus the definite integral of other [br]functions (say g(x)). Thus, it can be represented as the following:[br][b]Area between two curves = ∫[sub]a[/sub][sup]b [/sup][f(x)-g(x)]dx[/b]