Discover what magic we can derive when we take a derivative, which is the slope of the tangent line at any point on a curve.
The topic that is now known as "calculus" was really called "the calculus of differentials" when first devised by Newton (and Leibniz) roughly four hundred years ago. To Newton, differentials were infinitely small "changes" in numbers that previous mathematics didn't know what to do with. Think this has no relevence to you? Well how would you figure out how fast something is going *right* at this moment (you'd have to figure out the very, very small change in distance over an infinitely small change in time)? This tutorial gives a gentle introduction to the world of Newton and Leibniz.
We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!
We told you about the derivatives of many functions, but you might want proof that what we told you is actually true. That's what this tutorial tries to do!
CK-12 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and problem solving for teaching CK-12 Calculus Student Edition. The solution guide is available upon request.
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 1 covers functions, limits, derivatives, and integration
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 2 covers integration, differential equations, sequences and series, and parametric equations and polar coordinates.
Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Due to the comprehensive nature of the material, we are offering the book in three volumes for flexibility and efficiency. Volume 3 covers parametric equations and polar coordinates, vectors, functions of several variables, multiple integration, and second-order differential equations.
Professor Kingan’s motivation for writing her free Calculus I textbook was to help address the departments high failure rates in Calculus. Along with another CUNY initiative to offer Calculus workshops in advance of taking the course, Kingan’s concise textbook in Calculus I offers students inside and outside of CUNY an opportunity to prepare for Calculus I at their own pace. She also believes that by providing free access to this material she could help to overcome some of the inequity students experience when Calculus is not offered in their high school. The textbook was written specifically for this pilot project.
This problem involves the meaning of numbers found on labels. When the level of accuracy is not given we need to make assumptions based on how the information is reported. The goal of the task is to stimulate a conversation about rounding and about how to record numbers with an appropriate level of accuracy, tying in directly to the standard N-Q.3. It is therefore better suited for instruction than for assessment purposes.
Students apply their knowledge of linear regression and design to solve a real-world challenge to create a better packing solution for shipping cell phones. They use different materials, such as cardboard, fabric, plastic, and rubber bands to create new “composite material” packaging containers. Teams each create four prototypes made of the same materials and constructed in the same way, with the only difference being their weights, so each one is fabricated with a different amount of material. They test the three heavier prototype packages by dropping them from different heights to see how well they protect a piece of glass inside (similar in size to iPhone 6). Then students use linear regression to predict from what height they can drop the fourth/final prototype of known mass without the “phone” breaking. Success is not breaking the glass but not underestimating the height by too much either, which means using math to accurately predict the optimum drop height.
The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of that context. It can be used as either an assessment or a teaching task.
The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols.
The purpose of this task is to use finite geometric series to investigate an amazing mathematical object that might inspire students' curiosity. The Cantor Set is an example of a fractal.
This short video and interactive assessment activity is designed to teach second graders about capacity comparison problems with illustrations - word problems.
This short video and interactive assessment activity is designed to teach third graders an overview of capacity IIi (metric units).
This short video and interactive assessment activity is designed to teach fourth graders about finding the amount of water with illustrations and calculations (metric units).
This short video and interactive assessment activity is designed to teach second graders about capacity problems with illustrations - word problems.
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
In the task "Carbon 14 Dating'' the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died and, as this task shows, this is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant. The equation for the amount of Carbon 14 remaining in the preserved plant is in many ways simpler here, using 12 as a base.