- Tangent Lines to Graphs
- Linear Approximations
- Basic Linear Approximations
- Combining Basic Linear Approximations
- Review Problems
- Algebraic View of Linearization
- Applications of Linearization
- Differentiability, the Tangent Line-Linear Approximation
- Estimating a Function Value Using the Linear Approximation
- Determining an Inverse Function
- Tangent Line to a Graph
- Tangent Line to the Graph of an Exponential Function
- Implicit Differentiation and Tangent Line to a Graph
- Tangent Lines to a Hyperbola
- Tangents for an Implicit Function
- Tangent Lines to a Curve
- Hawk Chasing a Mouse
- Derivative and Tangent Line
- Constant, Linear, Quadratic, and Cubic Approximations
Tangent Lines to Graphs
Secant line and tangent line formally defined, with calculation example.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Linear Approximations
Linearizations of functions are defined with an example.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Definition and equation for the linear approximation of a function at a point. Includes diagrams and examples of basic linearizations.
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
Basic Linear Approximations
Several important linear approximations.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Combining Basic Linear Approximations
Linear approximations at x=a of f(x), f(c*x), c*f(x), (f+g)(x), f(x)g(x), f(x)/g(x), and f(g(x)). Includes derivations of each and a worked example.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Review Problems
Problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Algebraic View of Linearization
Relating linear approximation formulas to algebraic concepts of geometric series, the binomial theorem, and the sine function. Includes examples and a diagram for the sine function.
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
Applications of Linearization
Using linearization to find the mass of a body according to special relativity and the mass of a body as it moves away from the earth.
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
Differentiability, the Tangent Line-Linear Approximation
Definition, including differentials and an applet for graphing a function and its derivative.
18.013A Calculus with Applications, Spring 2005
Prof. Daniel J. Kleitman
Course Material Related to This Topic:
Estimating a Function Value Using the Linear Approximation
Estimating the values of an unknown function using linear approximation.
18.013A Calculus with Applications, Spring 2005
Prof. Daniel J. Kleitman
Course Material Related to This Topic:
Determining an Inverse Function
Method for using iterated linear approximations to find an inverse function.
18.013A Calculus with Applications, Spring 2005
Prof. Daniel J. Kleitman
Course Material Related to This Topic:
Tangent Line to a Graph
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
Three part question involving the tangent lines to the graph of f(x) = 1/x.
- Complete practice problems 1 on pages 1–2
- Check solution to practice problems 1 on pages 8–10
Finding a tangent line to the graph of a function through a point not on the graph.
- Complete exam problem 4 on page 5
- Check solution to exam problem 4 on pages 5–6
Finding the tangent lines to the graph of the exponential function through a point not on the graph.
- Complete exam problem 1.1 on page 1
- Check solution to exam problem 1.1 on page 1
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
Finding the equation in slope-intercept form of a line tangent to a graph at a given point.
- Complete exam problem 2 on page 1
- Check solution to
exam problem 2 on page 1
Tangent Line to the Graph of an Exponential Function
Finding the equation for the tangent line to an exponential function through a point not on the graph of the function.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
- Complete practice problems 1 on page 1
- Check solution to practice problems 1 on pages 6–7
Implicit Differentiation and Tangent Line to a Graph
Finding the equation of a tangent line to the graph of a function that is defined with an implicit equation.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
- Complete exam problem 3 on page 4
- Check solution to exam problem 3 on pages 4–5
Tangent Lines to a Hyperbola
Finding the lines through a given point which are tangent to a hyperbola.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
- Complete exam problem 11 on page 1
- Check solution to exam problem 11 on page 4
Tangents for an Implicit Function
Finding the tangent lines to the graph of a function defined with an implicit equation through a point not on the graph.
18.01 Single Variable Calculus, Fall 2005
Prof. Jason Starr
Course Material Related to This Topic:
- Complete exam problem 1.2 on page 1
- Check solution to exam problem 1.2 on page 1
Tangent Lines to a Curve
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
Two questions, one finding the horizontal tangent lines to a given curve and the other finding where a tangent line to a curve crosses the x-axis.
- Complete exam problems 6 to 7 on page 1
- Check solution to exam problems 6 to 7 on pages 4–5
Finding the tangent line to a curve at a point.
- Complete exam problem 1 on page 1
- Check solution to exam problem 1 on page 1
Hawk Chasing a Mouse
Using an exponential function to track the movement of a hawk as it chases a mouse.
18.01 Single Variable Calculus, Fall 2006
Prof. David Jerison
Course Material Related to This Topic:
- Complete exam problem 7 on page 1
- Check solution to exam problem 7 on page 1
Derivative and Tangent Line
Applet which plots the derivative of a function and shows the relationship between the derivative and the tangent line at a point.
18.013A Calculus with Applications, Spring 2005
Prof. Daniel J. Kleitman
Course Material Related to This Topic:
- Interact with a Java Simulation
Constant, Linear, Quadratic, and Cubic Approximations
Applet which will show some or all of these problems for a specified function at a chosen point.
18.013A Calculus with Applications, Spring 2005
Prof. Daniel J. Kleitman
Course Material Related to This Topic:
- Interact with a Java Simulation