MAT 1234 Calculus I Section 2.4 Derivatives of Tri. Functions

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Exam 1 Tutoring Record Bring it to class tomorrow! Get a new one for exam 2!

HW and Quiz WebAssign HW 2.4 Quiz: 2.3, 2.4

Preview Skills Formulas for the derivatives of tri. functions Find limits by change of variables Concepts Find limits by simple geometric insights an application of the squeeze theorem

Formulas

Why?

Formulas We are going to look at the first limit later.

Example 1

Example 2

Important Limit Use to find the formulas for the derivatives of the tri. functions Use to find other limits Use often in physics for approximations e.g. mechanical system, optics

Example Simple Pendulum When the angle is small, the motion can be modeled by

Important Limit Evidence: Graphs Proofs (a) Geometric proof (Section 2.4) (b) L’ hospital Rule (Section 6.8) (c) Taylor Series (Section 11.10) Why?

Important Limit Evidence: Graphs

Important Limit Evidence: Graphs

Important Limit Proofs (a) Geometric proof (Section 2.4) (b) L’ hospital Rule (Section 6.8) (c) Taylor Series (Section 11.10)

Example 3

Example 4

Generalized Formula Why?

Example 5

Remark It is incorrect to use the limit laws and write since we do not know the existence of

Example 6

Purposes (Skip if …) Look at the interesting power of geometry. Look at an application of the squeeze theorem.

Geometric Proof (Idea)

Simplified Proof:

Important Limit