Calculus i mat 145 dr day mon day march 4 2013
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Calculus I (MAT 145) Dr. Day Mon day March 4, 2013. Implicit Differentiation (3.5) Derivatives Involving Logarithms (3.6) Logarithmic Differentiation (3.6) Assignments. Using Derivative Patterns. For s(t) = cos (2t): Calculate s’(t) and s’’(t).

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Calculus I (MAT 145) Dr. Day Mon day March 4, 2013

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Calculus i mat 145 dr day mon day march 4 2013

Calculus I (MAT 145)Dr. DayMonday March 4, 2013

  • Implicit Differentiation (3.5)

  • Derivatives Involving Logarithms (3.6)

  • Logarithmic Differentiation (3.6)

  • Assignments

MAT 145


Using derivative patterns

Using Derivative Patterns

For s(t) = cos(2t):

  • Calculate s’(t)and s’’(t).

  • Determine an equation for the line tangent to the graph of s when t = π/8.

  • Determine the two values of t closest to t = 0 that lead to horizontal tangent lines.

  • Determine the smallest positive value of t for which s’(t) = 1.

  • If s(t) represents an object’s position on the number line at time t (s in feet, t in minutes), calculate the object’s velocity and acceleration at time t = π/12. Based on those results, describe everything you can about the object’s movement at that instant.

MAT 145


Derivatives of composite functions 3 4

Derivatives of Composite Functions (3.4)

MAT 145


Calculus i mat 145 dr day mon day march 4 2013

THE CHAIN RULE

WORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE"

Here's a function in a function

And your job here is to find

The derivative of the whole thing

With respect to x inside.

Call the outside f of u

And call the inside u of x.

Differentiate to find df/du

And multiply by du/dx.

Use the chain rule.

Use the chain rule.

Use the chain rule whene'er you find

The derivative of a function compositionally defined.

MAT 145


Derivatives of composite functions 3 41

Derivatives of Composite Functions (3.4)

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Derivatives of implicitly defined functions 3 5

Derivatives of Implicitly Defined Functions (3.5)

An implicitly defined function is:

A function whose relation among the variable is given by an equation for which the function has not been explicitly stated.

In the equation x2 + y2 = 25, y is an implicit function of x because the equation doesn’t explicitly express y in terms of x.

MAT 145


Derivatives of implicitly defined functions 3 51

Derivatives of Implicitly Defined Functions (3.5)

  • To calculate the derivative of an implicitly defined function:

  • Assume a functional connection among the variables. If x and y are present, assume y is a function of x.

  • Calculate the derivative of each term in the equation. Because we don’t explicitly know how y is determined by x, when calculating the derivative of y, we use the chain rule and write dy/dx as the derivative of y.

  • After determining term-by-term derivatives, carry out all necessary algebra steps to isolatedy/dx. That’s our goal!

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Derivatives of implicitly defined functions 3 52

Derivatives of Implicitly Defined Functions (3.5)

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Derivatives of implicitly defined functions 3 53

Derivatives of Implicitly Defined Functions (3.5)

Try These:

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Derivatives of logarithmic functions 3 6

Derivatives of Logarithmic Functions (3.6)

What is the derivative of the natural log function y = ln(x)?

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Derivatives of logarithmic functions 3 61

Derivatives of Logarithmic Functions (3.6)

Now apply this to other log functions:

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Derivatives of logarithmic functions 3 62

Derivatives of Logarithmic Functions (3.6)

And extend this to other functions:

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Assignments

Assignments

WebAssign

  • 3.4 (Part 2), 3.5, 3.6 (two parts) assignments up for completion.

  • Gateway Derivatives Quiz #2 returned today. GDQ #3 next week!

  • Test #3: Friday, March 8.

MAT 145


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