We told you about the derivatives of many functions, but you might …

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!

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CK-12 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and …

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.

The purpose of this task is to give students practice constructing functions …

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 …

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 task requires the student to use logarithms to solve an exponential …

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 …

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.

This problem introduces the method used by scientists to date certain organic …

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies. This problem is intended for instructional purposes only. It provides an interesting and important example of mathematical modeling with an exponential function.

This exploratory task requires the student to use a property of exponential …

This exploratory task requires the student to use a property of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

This simple task assesses whether students can interpret function notation. The four …

This simple task assesses whether students can interpret function notation. The four parts of the task provide a logical progression of exercises for advancing understanding of function notation and how to interpret it in terms of a given context.

This task is intended strictly for instructional purposes with the goal of …

This task is intended strictly for instructional purposes with the goal of building understandings of linear relationships within a meaningful and, hopefully, somewhat familiar context.

This task gives students an opportunity to work with exponential functions in …

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

This task develops reasoning behind the general formula for balances under continuously …

This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically addresses the standard (F-BF), building functions from a context, a auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks.

The purpose of this task is to introduce or reinforce the concept …

The purpose of this task is to introduce or reinforce the concept of a function, especially in a context where the function is not given by an explicit algebraic representation. Further, the last part of the task emphasizes the significance of one variable being a function of another variable in an immediately relevant real-life context.

In this visualization adapted from the University of Massachusetts Medical School, discover …

In this visualization adapted from the University of Massachusetts Medical School, discover the role that dengue viral proteins play in a human cell as the virus prepares to replicate.

This task asks students to find a linear function that models something …

This task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation.

This problem allows the student to think geometrically about lines and then …

This problem allows the student to think geometrically about lines and then relate this geometry to linear functions. Or the student can work algebraically with equations in order to find the explicit equation of the line through two points (when that line is not vertical).

This task is designed as a follow-up to the task F-LE Do …

This task is designed as a follow-up to the task F-LE Do Two Points Always Determine a Linear Function? Linear equations and linear functions are closely related, and there advantages and disadvantages to viewing a given problem through each of these points of view. This task is intended to show the depth of the standard F-LE.2 and its relationship to other important concepts of the middle school and high school curriculum, including ratio, algebra, and geometry.

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