calc ch 2 review

8/4/2019 Calc Ch 2 Review

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Problem: 9 Set: Review Exercises Page: 150

Look in your textbook for this problem statement.Step 1

Simplifying the function we get

Step 2

Differentiate each term using the power rule

Simplifying,



Expand the product to get

Combining terms,

Step 2

Differentiating, we get

Simplifying,


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Rewriting allows us to use the Generalized Power Rule, as follows

Therefore

Taking the derivatives and simplifying,

Finally,



Writing out the quotient rule, you get

Deriving,

Expanding the products in the numerator,


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Combining terms to simplify,



Rewriting the function

Taking the derivatives

Simplifying,

Problem: 23 Set: Review Exercises Page: 150Look in your textbook for this problem statement.

Step 1

Use the Chain Rule to get

Deriving,

Finally,



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Look in your textbook for this problem statement.

Step 1

Writing out the Chain Rule, we get

Deriving


Step 1

Differentiate each term. Use the Chain Rule on the second term

or

Therefore

Simplify using the DoubleAngle Formula for cos u



Differentiating term by term


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Differentiating while canceling the fractions

Step 2

Factor out

to get

Simplifying


Step 1

Use the product rule to get

Step 2

Simplify.



Use the Quotient Rule to obtain


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Taking the derivatives,

Dividing the numerator and denominator by x



Use the Chain Rule to get

Taking the derivative, you get

Step 2

Plug in x = 2/3 to get

Therefore

The calculator yields1.3603.


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Step 3

Graphing the function f(in blue) and f (in red), we see that the above answer does appear to becorrect.



Differentiating the function, term by term

Step 2

Deriving a second time, term by term



Step 1

Writing out the implicit derivative using the Chain and Generalized Power Rule


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Taking the derivatives,

or

Step 2

Isolating like terms on either side of the equation.

Step 3

Factor out the common multiple on the left hand side.

Finally,



Use the Product Rule and the Chain Rule to differentiate both sides of the equation with respectto x.


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Therefore,

Step 2

Isolating like terms on either side of the equation

Step 3

Factor out the common multiple on the left hand side

Finally,



Step 1

Differentiate to find the slope at x = 2 using implicit differentiation

or

Solving for dy/dx


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Evaluated at (2, 4),

Step 2

Substitute m =1/2, x1 = 2, and y1 = 4 into the PointSlope Equation of a Line

Therefore,

Step 3

Substitute m = 2,x1 = 2, and y1 = 4 into the PointSlope Equation of a Line.

or


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Step 1

Take the derivative of the function

Step 2

Set the derivative equal to1

Simplifying

or

Therefore

And the corresponding y values are


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Step 3

Set the derivative equal to 2

Simplifying

or

Therefore x = 1,3 and the corresponding y values2/3 and 2

Step 4

Set the derivative equal to 0

Therefore x = 0.414 and2.414 and the corresponding y values are1.219 and 2.552



Now we're left to determine if the function is differentiable at this point

Step 2

The function is continuous but not differentiable at x = 2


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Step 1

Take the derivative of the function twice

And

Step 2

Substitute into the given equation to get

Therefore, the given function satisfies the differential equation.

calc ch 2 review

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