ch.8: applications of derivatives (part...

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CH.8: APPLICATIONS OF DERIVATIVES (PART 1)

IMPLICIT DIFFERENTIATION

● Equations can be written in two forms: Explicitly and Implicitly.

● Example of Explicit form is ____________________

● Example of Implicit form is ____________________

EXAMPLE 1: Find dy/dx, Given:

A)

B)

C)

D)

Explicitly vs. Implicitly

Step Box

1. Derive both sides. 2. Apply dy/dx to all y. 3. Move terms with dy/dx to one side 4. Solve for dy/dx

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PRACTICE: IMPLICIT DIFFERENTIATION

PROBLEM: Derive the following function:

1.

2.

A

B

C

D

A

B

C

D

Step Box


Step Box


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PROBLEM: Derive the following function:

3.

4.

A

B

C

D

A

B

C

D

Step Box


Step Box


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PROBLEM: Derive the following function, SIMPLIFY your answer:

5.

A

B

C

D

Step Box


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PROBLEM: Derive the following function and SIMPLIFY your answer:

6.

A

B

C

D

Step Box


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RELATED RATES (Part 1) ● Used for finding an unknown rate of change, using a (known / unknown) rate of change. ● Derived in terms of _________________. ● Identify proper _________________ needed in problem. ● Identify the _________________ used in the equation. ● Be careful with _________________, use them to check your answer. EXAMPLE 1: The radius of a sphere is increasing at a rate of 3 m/s. How fast is the volume increasing when the diameter is 10 meters?

Equation:

Given:

EXAMPLE 2: A 5-ft latter is leaning against a vertical wall when Rosa begins pulling the bottom of the ladder away from the wall at a rate of .5 ft/s. How fast is the top of the ladder sliding down the wall when the foot of the ladder is 3-ft from the wall?

Equation:

Given:

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PRACTICE: RELATED RATES PROBLEM: Evaluate the following related rate. A 15 foot ladder is resting against the wall. The bottom is initially 10 feet away from the wall and is being pushed towards the wall at a rate of 1/4ft/sec. How fast is the top of the ladder moving up the wall 12 seconds after we start pushing?

Equation:

Given:

A 7√11176 𝑓𝑡

B −7√11176

𝑓𝑡𝑠

C 7√11176

𝑓𝑡𝑠

D 12√5

𝑓𝑡𝑠

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RELATED RATES (Part 2) ● Derived in terms of _________________. ● Identify proper _________________ needed in problem. ● Identify the _________________ used in the equation. ● Be careful with _________________, use them to check your answer. EXAMPLE 1: A cylindrical tank with radius 10 m is being filled with water at a rate of 2 m4 s⁄ . How fast is the height of the water increasing?

Equation:

Given:

EXAMPLE 2: Water is being poured into a conical reservoir at the rate of 9 pi cubic feet per second. The reservoir has a radius of 6 feet across the top and a height of 12 feet. At what rate is the depth of the water increasing when the depth is 6 feet?

Equation:

Given:

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PRACTICE: RELATED RATES PROBLEM: Evaluate the following related rate.

1) Each side of a square is increasing at a rate of 4m s⁄ . At what rate is the area of the square increasing when the area of the square is 16 m7 ?

Equation:

Given:

A 32𝑚 𝑠⁄

B −32𝑚 𝑠⁄

C 128𝑚 𝑠⁄

D −128𝑚 𝑠⁄

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MEAN VALUE THEOREM

● Mean Value Theorem is also referred to as ____________ .

●Conditions of the MVT:

o is continuous over the interval ________________

o is differentiable over the interval ______________

● There exists a point C When the slope of the tangent and the

slope of the secant are (different / equal / reciprocal)

EXAMPLE 1: Verify that the hypothesis of the Mean-Value Theorem is satisfied on the given interval. Find all the values of c

in the interval that satisfies the conclusion of the theorem.

EXAMPLE 2: Verify that the hypothesis of the Mean-Value Theorem is satisfied on the given interval. Find all the values of c

in that interval that satisfies the conclusion of the theorem.

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PRACTICE: MEAN VALUE THEOREM

1. Verify that the hypothesis of the Mean-Value Theorem is satisfied on the given interval. Find all the values of c in the

interval that satisfies the conclusion of the theorem.

( ) ( )

2. Verify that the hypothesis of the Mean-Value Theorem is satisfied on the given interval. Find all the values of c in the

interval that satisfies the conclusion of the theorem.

( )

⁄

A ( )

B

( )

C

( )

D

A

B

C

D

( ) ( ) ( )

( ) ( ) ( )

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ROLLE’S THEOREM

●Conditions of the Rolle’s Theorem:

o is continuous over the interval ________________

o is differentiable over the interval _______________

● Rolle’s theorem is applied to find a (horizontal / vertical ) tangent at c.

EXAMPLE 1: Find an interval on which that satisfies the hypothesis of the Rolle’s

Theorem.

EXAMPLE 2: Determine whether the Rolle’s Theorem can be applied. If so, find all values of c such that

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ch.8: applications of derivatives (part...

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