calc ch 2 review
TRANSCRIPT
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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,
Problem: 15 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
Expand the product to get
Combining terms,
Step 2
Differentiating, we get
Simplifying,
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Problem: 17 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
Rewriting allows us to use the Generalized Power Rule, as follows
Therefore
Taking the derivatives and simplifying,
Finally,
Problem: 19 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
Writing out the quotient rule, you get
Deriving,
Expanding the products in the numerator,
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Combining terms to simplify,
Problem: 21 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
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,
Problem: 25 Set: Review Exercises Page: 150
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Look in your textbook for this problem statement.
Step 1
Writing out the Chain Rule, we get
Deriving
Problem: 27 Set: Review Exercises Page: 150Look in your textbook for this problem statement.
Step 1
Differentiate each term. Use the Chain Rule on the second term
or
Therefore
Simplify using the DoubleAngle Formula for cos u
Problem: 29 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
Differentiating term by term
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Differentiating while canceling the fractions
Step 2
Factor out
to get
Simplifying
Problem: 31 Set: Review Exercises Page: 150Look in your textbook for this problem statement.
Step 1
Use the product rule to get
Step 2
Simplify.
Problem: 33 Set: Review Exercises Page: 150
Look in your textbook for this problem statement.Step 1
Use the Quotient Rule to obtain
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Taking the derivatives,
Dividing the numerator and denominator by x
Problem: 43 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.Step 1
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.
Problem: 45 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.Step 1
Differentiating the function, term by term
Step 2
Deriving a second time, term by term
Problem: 53 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.
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,
Problem: 57 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.Step 1
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,
Problem: 61 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.
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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Problem: 65 Set: Review Exercises Page: 151Look in your textbook for this problem statement.
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
Problem: 67 Set: Review Exercises Page: 151
Look in your textbook for this problem statement.Step 1
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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Problem: 69 Set: Review Exercises Page: 151Look in your textbook for this problem statement.
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.