As rates of change, derivatives give us information about the shape of a graph. In this course, we will apply the derivative to find linear approximations for single-variable and multi-variable functions. This gives us a straightforward way to estimate functions that may be complicated or difficult to evaluate. We will also use the derivative to locate the maximum and minimum values of a function. These optimization techniques are important for all fields, including the natural sciences and data analysis. The topics in this course lend themselves to many real-world applications, such as machine learning, minimizing costs or maximizing profits.
This course is part of the Differential Calculus through Data and Modeling Specialization
Offered By
Differential Calculus through Data and Modeling SpecializationJohns Hopkins UniversityAbout this Course
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Intermediate Level
Approx. 7 hours to complete
English
Subtitles: English
Shareable Certificate
Earn a Certificate upon completion
100% online
Start instantly and learn at your own schedule.
Course 4 of 4 in the
Flexible deadlines
Reset deadlines in accordance to your schedule.
Intermediate Level
Approx. 7 hours to complete
English
Subtitles: English
Offered by
Johns Hopkins University
The mission of The Johns Hopkins University is to educate its students and cultivate their capacity for life-long learning, to foster independent and original research, and to bring the benefits of discovery to the world.
Syllabus - What you will learn from this course
1 hour to complete
Linear Approximations and Tangent Planes
In single variable calculus, the derivative computes the slope of the tangent line where defined. This is then used to create the equation of the tangent line at a point, which can be used as an accurate estimation tool for complicated functions. This theory generalizes to lines in space which are used to create tangent planes. In this module, we work through the formulas and applications of these notions, using our developed theory of derivatives and partial derivatives.
1 hour to complete
2 videos (Total 35 min), 2 readings, 1 quiz
2 readings
Notes: Linear Approximation and Tangent Planes10m
Sample Problems: Linear Approximation and Tangent Planes10m
1 practice exercise
Linear Approximations and Tangent Planes30m
2 hours to complete
Maxima and Minima of Single-Variable Functions
Some of the most important applications of differential calculus are optimization problems in which the goal is to find the optimal (best) solution. For example, problems in marketing, economics, inventory analysis, machine learning, and business are all concerned with finding the best solution. These problems can be reduced to finding the maximum or minimum values of a function using our notions of the derivative.
2 hours to complete
2 videos (Total 49 min), 2 readings, 1 quiz
2 videos
How Derivatives Affect the Shape of a Graph23m
2 readings
Notes: Maxima and Minima of Single-Variable Functions10m
Sample Problems: Extrema of Single Variable Functions10m
1 practice exercise
Maxima and Minima of Single-Variable Functions30m
1 hour to complete
Maxima and Minima of Multivariable Functions
As models become more complicated, the functions used to describe them do as well. Many functions require more than one input to describe their output. These multivariable functions also contain maximum and minimum values that we seek to find using the tools of calculus. In this module, we will extend our optimization techniques to multivariable functions.
1 hour to complete
1 video (Total 24 min), 2 readings, 1 quiz
2 readings
Notes: Maxima and Minima of Multivariable Functions10m
Sample Problems: Extrema of Multivariable Functions10m
1 practice exercise
Maxima and Minima of Multivariable Functions30m
1 hour to complete
Lagrange Multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. It is named after the mathematician Joseph-Louis Lagrange. In this module, we develop the theory and work through examples of this powerful tool which converts a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a usually easier reformulation of the original problem.
1 hour to complete
1 video (Total 29 min), 2 readings, 1 quiz
1 video
2 readings
Notes: Lagrange Multipliers10m
Sample Problems: Lagrange Multipliers10m
1 practice exercise
Lagrange Multipliers30m
About the Differential Calculus through Data and Modeling Specialization
This specialization provides an introduction to topics in single and multivariable calculus, and focuses on using calculus to address questions in the natural and social sciences. Students will learn to use the tools of calculus to process, analyze, and interpret data, and to communicate meaningful results, using scientific computing and mathematical modeling. Topics include functions as models of data, differential and integral calculus of functions of one and several variables, differential equations, and optimization and estimation techniques.
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