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Boundless Lecture Slides

Boundless Lecture Slides. Available on the Boundless Teaching Platform. Free to share, print, make copies and changes. Get yours at www.boundless.com. Using Boundless Presentations. Boundless Teaching Platform

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Boundless Lecture Slides

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  1. Boundless Lecture Slides Available on the Boundless Teaching Platform Free to share, print, make copies and changes. Get yours at www.boundless.com

  2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

  3. About Boundless • Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com

  4. Precalculus Review Building Blocks of Calculus Functions and Models Limits ] Building Blocks of Calculus Free to share, print, make copies and changes. Get yours at www.boundless.com

  5. Building Blocks of Calculus > Precalculus Review Precalculus Review • Real Numbers, Functions, and Graphs • Linear and Quadratic Functions • Trigonometric Functions • Inverse Functions • Exponential and Logarithmic Functions • Using Calculators and Computers Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/building-blocks-of-calculus-1/precalculus-review-6/

  6. Building Blocks of Calculus > Functions and Models Functions and Models • Four Ways to Represent a Function • Essential Functions for Mathematical Modeling Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/building-blocks-of-calculus-1/functions-and-models-7/

  7. Building Blocks of Calculus > Limits Limits • Tangent and Velocity Problems • Limit of a Function • Calculating Limits Using the Limit Laws • Precise Definition of a Limit • Continuity • Finding Limits Algebraically • Trigonometric Limits • Intermediate Value Theorem • Infinite Limits Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/building-blocks-of-calculus-1/limits-8/

  8. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

  9. Building Blocks of Calculus Key terms • algebraiccontaining only numbers, letters, and arithmetic operators • asymptotea straight line which a curve approaches arbitrarily closely, as they go to infinity • bicontinuoushomomorphic or of structure-preserving mapping • binarythe bijective base-2 numeral system, which uses only the digits 0 and 1 • Cartesianof or pertaining to coordinates based on mutually orthogonal axes • completeness of the real numberscompleteness implies that there are not any "gaps" or "missing points" in the real number line • continuous functiona function whose value changes only slightly when its input changes slightly • definitiona formalization of the notion of the limit of functions • derivativea measure of how a function changes as its input changes • derivativea measure of how a function changes as its input changes • derivativea measure of how a function changes as its input changes • differentiablehaving a derivative, said of a function whose domain and co-domain are manifolds Free to share, print, make copies and changes. Get yours at www.boundless.com

  10. Building Blocks of Calculus • differential equationan equation involving the derivatives of a function • domainthe set of all possible mathematical entities (points) where a given function is defined • domainthe set of all possible mathematical entities (points) where a given function is defined • errorthe difference between a measured or calculated value and a true one • exponential growthThe growth in the value of a quantity, in which the rate of growth is proportional to the instantaneous value of the quantity; for example, when the value has doubled, the rate of increase will also have doubled. The rate may be positive or negative. • functiona relation in which each element of the domain is associated with exactly one element of the co-domain • functiona relation in which each element of the domain is associated with exactly one element of the co-domain • hypotenusethe side of a right triangle opposite the right angle • infinitya number that has an infinite numerical value that cannot be counted • injectiveof, relating to, or being an injection: such that each element of the image (or range) is associated with at most one element of the preimage (or domain); inverse-deterministic • integralalso sometimes called antiderivative; the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed • inverse functiona function that does exactly the opposite of another Free to share, print, make copies and changes. Get yours at www.boundless.com

  11. Building Blocks of Calculus • limita value to which a sequence or function converges • linearitya relationship between several quantities which can be considered proportional and expressed in terms of linear algebra; any mathematical property of a relationship, operation, or function that is analogous to such proportionality, satisfying additivity and homogeneity • map projectionany systematic method of transforming the spherical representation of parallels, meridians and geographic features of the Earth's surface to a nonspherical surface, usually a plane • mathematical modelAn abstract mathematical representation of a process, device or concept; it uses a number of variables to represent inputs, outputs and internal states, and sets of equations and inequalities to describe their interaction. • polynomialan expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power • programming languagecode of reserved words and symbols used in computer programs, which give instructions to the computer on how to accomplish certain computing tasks • rangethe set of values (points) which a function can obtain • real numbera value that represents a quantity along a continuous line • secant linea line that (locally) intersects two points on the curve • squeeze theoremtheorem obtaining the limit of a function via comparison with two other functions whose limits are known or easily computed • supercomputerany computer that has a far greater processing power than others of its day; typically they use more than one core and are housed in large clean rooms with high air flow to permit cooling. Typical uses are weather forecasting, nuclear simulations and animations • tangenta straight line touching a curve at a single point without crossing it there Free to share, print, make copies and changes. Get yours at www.boundless.com

  12. Building Blocks of Calculus • topologya branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms • trigonometric functionany function of an angle expressed as the ratio of two of the sides of a right triangle that has that angle, or various other functions that subtract 1 from this value or subtract this value from 1 (such as the versed sine) • velocitya vector quantity that denotes the rate of change of position with respect to time, or a speed with the directional component Free to share, print, make copies and changes. Get yours at www.boundless.com

  13. Building Blocks of Calculus Fig 1 A typical graphing calculator built by Texas Instruments, displaying a graph of a function . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Scientific calculator."CC BYhttp://en.wikipedia.org/wiki/Scientific_calculatorView on Boundless.com

  14. Building Blocks of Calculus Finding a Limit The limit of as goes to infinity can be segmented down into two parts: the limit of and the limit of . The former is , while the latter is . Therefore, the limit of as goes to infinity is . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Limit of a function."CC BYhttp://en.wikipedia.org/wiki/Limit_of_a_function%23PropertiesView on Boundless.com

  15. Building Blocks of Calculus Graph of a Function This is a graph of the function Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Function (mathematics)."CC BYhttp://en.wikipedia.org/wiki/Function_(mathematics)View on Boundless.com

  16. Building Blocks of Calculus Right-Angle Triangle Right-angle triangle used in defining trigonometric functions. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Trigonometric functions."CC BYhttp://en.wikipedia.org/wiki/Trigonometric_functionsView on Boundless.com

  17. Building Blocks of Calculus A Function and its Inverse A function and its inverse . Because maps to 3, the inverse maps 3 back to . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Inverse function."CC BYhttp://en.wikipedia.org/wiki/Inverse_functionView on Boundless.com

  18. Building Blocks of Calculus Graph of The derivative (or slope of a tangential line) of the exponential function is equal to the value of the function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential function."CC BYhttp://en.wikipedia.org/wiki/Exponential_functionView on Boundless.com

  19. Building Blocks of Calculus Fig 2 A three-dimentional wireframe plot of the unnormalized sinc function . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Matlab."CC BYhttp://en.wikipedia.org/wiki/MatlabView on Boundless.com

  20. Building Blocks of Calculus Plot of The graph of the logarithm to base 2 crosses the x-axis (horizontal axis) at 1 and passes through the points with coordinates , , and . For example, , because . The graph gets arbitrarily close to the y-axis, but does not meet or intersect it. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Binary_logarithm_plot_with_ticks.png."CC BY-SA 4.0https://en.wikipedia.org/wiki/LogarithmView on Boundless.com

  21. Building Blocks of Calculus Essential Discontinuity A graph of the above function, demonstrating that the limit at does not exist. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Discontinuity_essential.eps.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/Limit_of_a_functionView on Boundless.com

  22. Building Blocks of Calculus Instantaneous Velocity The green line shows the tangential line of the position-time curve at a particular time. Its slope is the velocity at that point. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Motion graphs and derivatives."CC BYhttp://en.wikipedia.org/wiki/Motion_graphs_and_derivativesView on Boundless.com

  23. Building Blocks of Calculus Definitely of a Limit Whenever a point is within units of , is within units of . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."(ε, δ)-definition of limit."CC BYhttp://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limitView on Boundless.com

  24. Building Blocks of Calculus Sine, Tangent, and Secant The sine, tangent, and secant functions of an angle constructed geometrically in terms of a unit circle. The number θ is the length of the curve; thus angles are being measured in radians. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Trigonometric functions."CC BYhttp://en.wikipedia.org/wiki/Trigonometric_functionsView on Boundless.com

  25. Building Blocks of Calculus Base of Trigonometry If two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: , ,  where  is the common measure of five acute angles. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Trigonometric functions."CC BYhttp://en.wikipedia.org/wiki/Trigonometric_functionsView on Boundless.com

  26. Building Blocks of Calculus Squeeze Theorem being squeezed by and in the limit as  approaches . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Squeeze theorem."CC BYhttp://en.wikipedia.org/wiki/Squeeze_theoremView on Boundless.com

  27. Building Blocks of Calculus Infinite Limit For any arbitrarily small , there exists a large enough such that when , . Therefore, the limit of this function at infinity exists. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Limit of a function."CC BYhttp://en.wikipedia.org/wiki/Limit_of_a_function%23Limits_involving_infinityView on Boundless.com

  28. Building Blocks of Calculus Limit of a Function at Infinity For an arbitrarily small , there always exists a large enough number such that when approaches , . Therefore, the limit of this function at infinity exists. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Limit of a function."CC BYhttp://en.wikipedia.org/wiki/Limit_of_a_functionView on Boundless.com

  29. Building Blocks of Calculus Nonexistence of Limit The limit as the function approaches from the left does not equal the limit as the function approaches from the right, so the limit of the function at does not exist. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Upper_semi.svg."CC BY-SA 3.0https://en.wikipedia.org/wiki/Limit_of_a_functionView on Boundless.com

  30. Building Blocks of Calculus Sinc Function The normalized sinc (blue, higher frequency) and unnormalized sinc function (red, lower frequency) shown on the same scale. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Si_sinc.svg."CC BY-SA 3.0https://en.wikipedia.org/wiki/Sinc_functionView on Boundless.com

  31. Building Blocks of Calculus Logistic Curve The standard logistic curve. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Logistic-curve.svg."CC BY-SA 4.0https://en.wikipedia.org/wiki/Logistic_functionView on Boundless.com

  32. Building Blocks of Calculus Exponential Growth The graph illustrates how exponential growth (green) surpasses both linear (red) and cubic (blue) growth. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Exponential.svg."CC BY-SA 4.0https://en.wikipedia.org/wiki/Exponential_growthView on Boundless.com

  33. Building Blocks of Calculus Attribution • Wikipedia."Function (mathematics)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Function_(mathematics) • Wikipedia."Real number."CC BY-SA 3.0http://en.wikipedia.org/wiki/Real_number • Wiktionary."domain."CC BY-SA 3.0http://en.wiktionary.org/wiki/domain • Wikipedia."Graph of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Graph_of_a_function • Wiktionary."Cartesian."CC BY-SA 3.0http://en.wiktionary.org/wiki/Cartesian • Wikipedia."Quadratic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Quadratic_function • Wikipedia."Linear function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Linear_function • Wiktionary."polynomial."CC BY-SA 3.0http://en.wiktionary.org/wiki/polynomial • Wikipedia."Affine function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Affine_function • Wiktionary."linearity."CC BY-SA 3.0http://en.wiktionary.org/wiki/linearity • Wikipedia."Trigonometric functions."CC BY-SA 3.0http://en.wikipedia.org/wiki/Trigonometric_functions • Wiktionary."differential equation."CC BY-SA 3.0http://en.wiktionary.org/wiki/differential_equation • Wiktionary."hypotenuse."CC BY-SA 3.0http://en.wiktionary.org/wiki/hypotenuse • Wikipedia."Inverse function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Inverse_function • Wiktionary."range."CC BY-SA 3.0http://en.wiktionary.org/wiki/range • Wiktionary."injective."CC BY-SA 3.0http://en.wiktionary.org/wiki/injective • Wiktionary."domain."CC BY-SA 3.0http://en.wiktionary.org/wiki/domain Free to share, print, make copies and changes. Get yours at www.boundless.com

  34. Building Blocks of Calculus • Wikipedia."Exponential function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Exponential_function • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wikipedia."Logarithm."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logarithm • Wiktionary."binary."CC BY-SA 3.0http://en.wiktionary.org/wiki/binary • Wiktionary."inverse function."CC BY-SA 3.0http://en.wiktionary.org/wiki/inverse_function • Wikipedia."Scientific calculator."CC BY-SA 3.0http://en.wikipedia.org/wiki/Scientific_calculator • Wiktionary."supercomputer."CC BY-SA 3.0http://en.wiktionary.org/wiki/supercomputer • Wiktionary."programming language."CC BY-SA 3.0http://en.wiktionary.org/wiki/programming_language • Wikipedia."Function (mathematics)."CC BY-SA 3.0http://en.wikipedia.org/wiki/Function_(mathematics) • Wiktionary."function."CC BY-SA 3.0http://en.wiktionary.org/wiki/function • OpenStax CNX."Kenny Felder, Function Concepts -- Four Ways to Represent a Function. September 17, 2013."CC BY 3.0http://cnx.org/content/m18195/latest/ • Wikipedia."Logistic function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Logistic_function • Wikipedia."Derivative."CC BY-SA 3.0https://en.wikipedia.org/wiki/Derivative • Wiktionary."mathematical model."CC BY-SA 3.0http://en.wiktionary.org/wiki/mathematical_model • Wikipedia."Mathematical model."CC BY-SA 3.0http://en.wikipedia.org/wiki/Mathematical_model • Wiktionary."map projection."CC BY-SA 3.0http://en.wiktionary.org/wiki/map_projection • Wikipedia."Logarithm."CC BY-SA 3.0https://en.wikipedia.org/wiki/Logarithm • Wikipedia."Malthusian growth model."CC BY-SA 3.0http://en.wikipedia.org/wiki/Malthusian_growth_model • Wiktionary."exponential growth."CC BY-SA 3.0http://en.wiktionary.org/wiki/exponential_growth Free to share, print, make copies and changes. Get yours at www.boundless.com

  35. Building Blocks of Calculus • Wikipedia."Velocity."CC BY-SA 3.0https://en.wikipedia.org/wiki/Velocity • Wiktionary."velocity."CC BY-SA 3.0http://en.wiktionary.org/wiki/velocity • Wiktionary."tangent."CC BY-SA 3.0http://en.wiktionary.org/wiki/tangent • Wiktionary."integral."CC BY-SA 3.0http://en.wiktionary.org/wiki/integral • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wiktionary."function."CC BY-SA 3.0http://en.wiktionary.org/wiki/function • Wikipedia."secant line."CC BY-SA 3.0http://en.wikipedia.org/wiki/secant%20line • Wikipedia."Limit of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_of_a_function • Wikipedia."derivative."CC BY-SA 3.0http://en.wikipedia.org/wiki/derivative • Wikipedia."Squeeze theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Squeeze_theorem • Wikipedia."L'Hôpital's rule."CC BY-SA 3.0http://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule • Wiktionary."differentiable."CC BY-SA 3.0http://en.wiktionary.org/wiki/differentiable • Wikipedia."(ε, δ)-definition of limit."CC BY-SA 3.0http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit • Wiktionary."infinity."CC BY-SA 3.0http://en.wiktionary.org/wiki/infinity • Wiktionary."error."CC BY-SA 3.0http://en.wiktionary.org/wiki/error • Wikipedia."Limit of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_of_a_function • Wikipedia."Continuous function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Continuous_function • Wiktionary."bicontinuous."CC BY-SA 3.0http://en.wiktionary.org/wiki/bicontinuous • Wiktionary."topology."CC BY-SA 3.0http://en.wiktionary.org/wiki/topology Free to share, print, make copies and changes. Get yours at www.boundless.com

  36. Building Blocks of Calculus • Wiktionary."algebraic."CC BY-SA 3.0http://en.wiktionary.org/wiki/algebraic • Wikipedia."Limit of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_of_a_function%23Properties • Wiktionary."limit."CC BY-SA 3.0http://en.wiktionary.org/wiki/limit • Wikipedia."squeeze theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/squeeze%20theorem • Wiktionary."trigonometric function."CC BY-SA 3.0http://en.wiktionary.org/wiki/trigonometric_function • Wikipedia."Limit of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_of_a_function%23Properties • Wikipedia."Intermediate value theorem."CC BY-SA 3.0http://en.wikipedia.org/wiki/Intermediate_value_theorem • Wiktionary."continuous function."CC BY-SA 3.0http://en.wiktionary.org/wiki/continuous_function • Wikipedia."completeness of the real numbers."CC BY-SA 3.0http://en.wikipedia.org/wiki/completeness%20of%20the%20real%20numbers • Wiktionary."real number."CC BY-SA 3.0http://en.wiktionary.org/wiki/real_number • Wiktionary."asymptote."CC BY-SA 3.0http://en.wiktionary.org/wiki/asymptote • Wikipedia."Limit of a function."CC BY-SA 3.0http://en.wikipedia.org/wiki/Limit_of_a_function%23Limits_involving_infinity • Wiktionary."definition."CC BY-SA 3.0http://en.wiktionary.org/wiki/definition Free to share, print, make copies and changes. Get yours at www.boundless.com

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