- 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

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:**

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:**

Several important linear approximations.

18.01

*Single Variable Calculus,*Fall 2005

Prof. Jason Starr

**Course Material Related to This Topic:**

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:**

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:**

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:**

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:**

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 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:**

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:**

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

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

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

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

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

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

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

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

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