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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. Derivatives Derivatives and Integrals Applications of Differentiation Integrals Applications of Integration ] Derivatives and Integrals Free to share, print, make copies and changes. Get yours at www.boundless.com

  5. Derivatives and Integrals > Derivatives Derivatives • The Derivative and Tangent Line Problem • Derivatives and Rates of Change • The Derivative as a Function • Differentiation Rules • Derivatives of Trigonometric Functions • The Chain Rule • Implicit Differentiation • Differentiation and Rates of Change in the Natural and Social Sciences • Related Rates • Higher Derivatives Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/derivatives-and-integrals-2/derivatives-9/

  6. Derivatives and Integrals > Applications of Differentiation Applications of Differentiation • Linear Approximation • Maximum and Minimum Values • The Mean Value Theorem, Rolle's Theorem, and Monotonicity • Derivatives and the Shape of the Graph • Horizontal Asymptotes and Limits at Infinity • Curve Sketching • Graphing on Computers and Calculators • Optimization • Newton's Method • Concavity and the Second Derivative Test • Differentials Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/derivatives-and-integrals-2/applications-of-differentiation-10/

  7. Derivatives and Integrals > Integrals Integrals • Antiderivatives • Area and Distances • The Definite Integral • The Fundamental Theorem of Calculus • Indefinite Integrals and the Net Change Theorem • The Substitution Rule • Further Transcendental Functions • Numerical Integration Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/derivatives-and-integrals-2/integrals-11/

  8. Derivatives and Integrals > Applications of Integration Applications of Integration • Area Between Curves • Volumes • Average Value of a Function • Cylindrical Shells • Work • Volumes of Revolution Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/calculus/textbooks/boundless-calculus-textbook/derivatives-and-integrals-2/applications-of-integration-12/

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

  10. Derivatives and Integrals Key terms • antiderivativean indefinite integral • antiderivativean indefinite integral • antiderivativean indefinite integral • antiderivativean indefinite integral • antiderivativean indefinite integral • approximationAn imprecise solution or result that is adequate for a defined purpose. • arctangentAny of several single-valued or multivalued functions that are inverses of the tangent function. • areaa measure of the extent of a surface measured in square units • arithmetic meanthe measure of central tendency of a set of values, computed by dividing the sum of the values by their number; commonly called the mean or the average • asymptotea straight line which a curve approaches arbitrarily closely, as they go to infinity • averageany measure of central tendency, especially any mean, median, or mode • axisa fixed, one-dimensional figure, such as a line or arc, with an origin and orientation and such that its points are in one-to-one correspondence with a set of numbers; an axis forms part of the basis of a space or is used to position and locate data in a graph (a coordinate axis) Free to share, print, make copies and changes. Get yours at www.boundless.com

  11. Derivatives and Integrals • compositea function of a function • concavecurved like the inner surface of a sphere or bowl • convexcurved or bowed outward like the outside of a bowl or sphere or circle • critical pointa maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined • critical pointa maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined • cuboida parallelepiped having six rectangular faces • curvea simple figure containing no straight portions and no angles • cylindera surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve • cylindera surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve • definite integralthe integral of a function between an upper and lower limit • definite integralthe integral of a function between an upper and lower limit • definite integralthe integral of a function between an upper and lower limit Free to share, print, make copies and changes. Get yours at www.boundless.com

  12. Derivatives and Integrals • definite integralthe integral of a function between an upper and lower limit • definite integralthe integral of a function between an upper and lower limit • definite integralthe integral of a function between an upper and lower limit • 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 • derivativea measure of how a function changes as its input changes • difference quotientthe function difference divided by the point difference : • differentiablehaving a derivative, said of a function whose domain and co-domain are manifolds • differential geometrythe study of geometry using differential calculus • domainthe set of all possible mathematical entities (points) where a given function is defined • exponential functionany function in which an independent variable is in the form of an exponent; they are the inverse functions of logarithms Free to share, print, make copies and changes. Get yours at www.boundless.com

  13. Derivatives and Integrals • forcea physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in  (SI: newton, abbreviated N; CGS: dyne, abbreviated dyn) • functiona relation in which each element of the domain is associated with exactly one element of the co-domain • graphA diagram displaying data; in particular one showing the relationship between two or more quantities, measurements or indicative numbers that may or may not have a specific mathematical formula relating them to each other. • gross domestic producta measure of the economic production of a particular territory in financial capital terms over a specific time period • implicitimplied indirectly, without being directly expressed • infinitesimala non-zero quantity whose magnitude is smaller than any positive number • 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 • 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 • integrationthe operation of finding the region in the -plane bound by the function • integrationthe operation of finding the region in the -plane bound by a given function • integrationthe operation of finding the region in the - plane bound by the function • integrationthe operation of finding the region in the -plane bound by the function Free to share, print, make copies and changes. Get yours at www.boundless.com

  14. Derivatives and Integrals • limita value to which a sequence or function converges • linearhaving the form of a line; straight • local maximumA maximum within a restricted domain, especially a point on a function whose value is greater than the values of all other points near it. • local minimumA point on a graph (or its associated function) such that the points each side have a greater value even though another point exists with a smaller value. • meanThe average value. • momentum(of a body in motion) the product of its mass and velocity • optimizationthe design and operation of a system or process to make it as good as possible in some defined sense • 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 • 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 • proprietaryManufactured exclusively by the owner of intellectual property rights (IPR), as with a patent or trade secret. • rangethe set of values (points) which a function can obtain • revolutionthe turning of an object about an axis Free to share, print, make copies and changes. Get yours at www.boundless.com

  15. Derivatives and Integrals • revolutionrotation: the turning of an object around an axis • rootA zero (of a function). • scientific calculatorAn electronic calculator that can handle trigonometric, exponential and often other advanced functions, and can show its output in scientific notation and sometimes in hexadecimal, octal or binary • secanta line that intersects a curve at two or more points • secanta straight line that intersects a curve at two or more points • secanta straight line that intersects a curve at two or more points • slopealso called gradient; slope or gradient of a line describes its steepness • spring constanta characteristic of a spring which is defined as the ratio of the force affecting the spring to the displacement caused by it • stochasticRandom, randomly determined • symmetryExact correspondence on either side of a dividing line, plane, center or axis. • tangenta line touching a curve at a single point without crossing it there • 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

  16. Derivatives and Integrals • tangenta straight line touching a curve at a single point without crossing it there • 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) • variablea quantity that may assume any one of a set of values • volumea unit of three-dimensional measure of space that comprises a length, a width and a height; measured in units of cubic centimeters in metric, cubic inches, or cubic feet in English measurement • volumea unit of three-dimensional measure of space that comprises a length, a width and a height; measured in units of cubic centimeters in metric, cubic inches, or cubic feet in English measurement Free to share, print, make copies and changes. Get yours at www.boundless.com

  17. Derivatives and Integrals Skydiving The path of a skydiver relies on many variables such as time and height. Use of the chain rule is needed for the complicated calculation. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Skydiving."CC BY-SAhttps://en.wikipedia.org/wiki/SkydivingView on Boundless.com

  18. Derivatives and Integrals Flow Chart for Related Rate Problem Solving Related rate problems cab be handled by taking a methodical approach. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Related rates."Public domainhttps://en.wikipedia.org/wiki/Related_ratesView on Boundless.com

  19. Derivatives and Integrals Maxima Finding maxima is useful in optimization problems. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Mathematical optimization."CC BYhttp://en.wikipedia.org/wiki/Mathematical_optimizationView on Boundless.com

  20. Derivatives and Integrals Integration: Area Under a Curve Integration can be thought of as measuring the area under a curve, defined by , between two points (here, and ). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Area."CC BYhttp://en.wikipedia.org/wiki/AreaView on Boundless.com

  21. Derivatives and Integrals Definite Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/420px-Integral_example.svg.pngView on Boundless.com

  22. Derivatives and Integrals Slope of a function A function with the slope shown for a given point. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Derivative."Public domainhttps://en.wikipedia.org/wiki/DerivativeView on Boundless.com

  23. Derivatives and Integrals Acceleration Acceleration is the time-rate of change of velocity, and the second-order rate of change of position. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Acceleration."CC BY-SAhttps://en.wikipedia.org/wiki/AccelerationView on Boundless.com

  24. Derivatives and Integrals Derivative At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where a green line appears, negative where a red line appears, and zero where a black line appears. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Derivative."CC BYhttps://en.wikipedia.org/wiki/DerivativeView on Boundless.com

  25. Derivatives and Integrals Tangent to a Curve The line shows the tangent to the curve at the point represented by the dot. It barely touches the curve and shows the rate of change slope at the point. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Tangent."Public domainhttps://en.wikipedia.org/wiki/TangentView on Boundless.com

  26. Derivatives and Integrals Path of a Point on a Circle The path of a point on a circle can only be expressed as an implicit function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Implicit function theorem."CC BY-SAhttps://en.wikipedia.org/wiki/Implicit_function_theoremView on Boundless.com

  27. Derivatives and Integrals Derivative As Slope The slope of tangent line shown represents the value of the derivative of the curved function at the point . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Tangent-calculus.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/DerivativeView on Boundless.com

  28. Derivatives and Integrals Discontinuous Function At the point where the function makes a jump, the derivative of the function does not exist. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Right-continuous.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/DerivativeView on Boundless.com

  29. Derivatives and Integrals Model Rockets The flight of model rockets can be modeled using the product rule. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikibooks."Calculus/Product and Quotient Rules."CC BY-SA 3.0http://en.wikibooks.org/wiki/Calculus/Product_and_Quotient_RulesView on Boundless.com

  30. Derivatives and Integrals Sine and Cosine In this image, one can see that where the line tangent to one curve has zero slope (the derivative of that curve is zero), the value of the other function is zero. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Sine_cosine_one_period.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/SineView on Boundless.com

  31. Derivatives and Integrals Differentials The differential of a function at a point . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Differentials."CC BYhttp://en.wikipedia.org/wiki/DifferentialsView on Boundless.com

  32. Derivatives and Integrals Maxima and Minima Telling whether a critical point is a maximum or a minimum has to do with the second derivative. If it is concave-up at the point, it is a minimum; if concave-down, it is a maximum. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Maxima and minima."CC BYhttp://en.wikipedia.org/wiki/Maxima_and_minimaView on Boundless.com

  33. Derivatives and Integrals Graphing Calculator Calculators graph curves by drawing each pixel as a linear approximation of the function. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Graphing calculator."CC BYhttp://en.wikipedia.org/wiki/Graphing_calculatorView on Boundless.com

  34. Derivatives and Integrals GraphCalc Screenshot of GraphCalc Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/d/d9/Graphcalc_screenshot.pngView on Boundless.com

  35. Derivatives and Integrals Newton's Method The function is shown in blue and the tangent line in red. We see that  is a better approximation than for the root of the function . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Newton's method."CC BYhttps://en.wikipedia.org/wiki/Newton's_methodView on Boundless.com

  36. Derivatives and Integrals Trapezoidal Rule Illustration of the trapezoidal rule. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Numerical integration."CC BYhttp://en.wikipedia.org/wiki/Numerical_integrationView on Boundless.com

  37. Derivatives and Integrals Definite Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/420px-Integral_example.svg.pngView on Boundless.com

  38. Derivatives and Integrals Definite Integral A definite integral of a function can be represented as the signed area of the region bounded by its graph. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Integral_example.svg/420px-Integral_example.svg.pngView on Boundless.com

  39. Derivatives and Integrals Rectangle Rule Illustration of the rectangle rule of numerical integration. The value of is taken to be constant around a point and the integral is calculated by adding up the areas of the rectangles. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Numerical integration."CC BYhttp://en.wikipedia.org/wiki/Numerical_integrationView on Boundless.com

  40. Derivatives and Integrals The Shell Method Calculating volume using the shell method. Each segment located at , between and the -axis, gives a cylindrical shell after revolution around the vertical axis. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Shell method."CC BYhttp://en.wikipedia.org/wiki/Shell_methodView on Boundless.com

  41. Derivatives and Integrals Trigonometric Functions Top panel: Trigonometric function sinθ for selected angles , , , and in the four quadrants. Bottom panel: Graph of sine function versus angle. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttps://upload.wikimedia.org/wikipedia/commons/0/08/Periodic_sine.PNGView on Boundless.com

  42. Derivatives and Integrals Area Between Two Graphs The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Area."CC BYhttp://en.wikipedia.org/wiki/AreaView on Boundless.com

  43. Derivatives and Integrals A Volume of Revolution A solid formed by rotating a curve around an axis. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Solid of revolution."CC BYhttp://en.wikipedia.org/wiki/Solid_of_revolutionView on Boundless.com

  44. Derivatives and Integrals Disc Integration Disc integration about the -axis. Integration is along the axis of revolution (-axis in this case). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Solid of revolution."CC BYhttp://en.wikipedia.org/wiki/Solid_of_revolutionView on Boundless.com

  45. Derivatives and Integrals Shell Integration The integration (along the -axis) is perpendicular to the axis of revolution (-axis). Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Solid of revolution."CC BYhttp://en.wikipedia.org/wiki/Solid_of_revolutionView on Boundless.com

  46. Derivatives and Integrals Fig 1 Triple integral of a constant function over the shaded region gives the volume. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Integral calculus."CC BYhttp://en.wikipedia.org/wiki/Integral_calculusView on Boundless.com

  47. Derivatives and Integrals Fig 1 The average of a function that has area over the range  is . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Area."CC BYhttp://en.wikipedia.org/wiki/AreaView on Boundless.com

  48. Derivatives and Integrals Spring and Restoring Force The spring applies a restoring force () on the object located at . Work done by the restoring force leads to increase in the kinetic energy of the object. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax CNX."Sunil Kumar Singh, Work by Spring Force. April 14, 2013."CC BY 3.0http://cnx.org/content/m14102/latest/View on Boundless.com

  49. Derivatives and Integrals The Fundamental Theorem of Calculus We can see from this picture that the Fundamental Theorem of Calculus works. By definition, the derivative of is equal to as tends to zero. By replacing the numerator, , by and dividing by , is obtained. Taking the limit as tends to zero completes the proof of the Fundamental Theorem of Calculus. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikimedia.CC BY-SAhttp://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/FTC_geometric.svg/627px-FTC_geometric.svg.pngView on Boundless.com

  50. Derivatives and Integrals Maxima and Minima Local and global maxima and minima for , . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Maxima and minima."CC BYhttp://en.wikipedia.org/wiki/Maxima_and_minimaView on Boundless.com

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