Compilation of Solved Problems in Differential Calculus

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  • Topic: Maxima and minima, Second derivative test, First derivative test
  • Pages : 14 (2506 words )
  • Download(s) : 129
  • Published : March 9, 2013
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Maxima and Minima

First Derivative Test

1) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', 4 is the only critical number of f. Testing:
x < 4| f'(0) = -8| f is decreasing|
x > 4| f'(5) = 2| f is increasing|
By the First Derivative Test, x = 4 is a local minimum.

2) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', -3 and 6 are the only critical numbers of f. Testing:
x < -3| f'(-10) = 672| f is increasing|
-3 < x < 6| f(0) = -108| f is decreasing|
x >6| f(10) = 312| f is increasing|
By the First Derivative Test, x = -3 is a local minimum and x = 6 is a local maximum.

3) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', 0 and 0.75 are the only critical numbers of f. Testing:
x < 0| f'(-1) = -7| f is decreasing|
0 < x < 0.75| f(0.5) = -0.25| f is decreasing|
x >0.75| f(1) = 1| f is increasing|
By the First Derivative Test, x = 1 is a local minimum.

4) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', -2 and 0 are the only critical numbers of f. Testing:
x < -2| f'(-10) = 80 e- 10| f is increasing|
-2 < x < 0| f(-1) = - e- 1| f is decreasing|
x > 0| f(1) = 3 e| f is increasing|
By the First Derivative Test, x = - 2 is a local minimum and x = 0 is a local maximum.

5)We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', 0 is the only critical number of f. Testing:
x < 0| f'(-1) = 8 e- 1| f is increasing|
x > 0| f'(1) = - 8 e- 1| f is decreasing|
By the First Derivative Test, x = 0 is a local maximum.

6) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since f' is not defined at - 3 and 3 which are in the domain of f, -3, 0 and 3 are the critical numbers of f.

Testing:
x < -3| f'(-10) = -20| f is decreasing|
-3 < x < 0| f(-1) = 2| f is increasing|
0 < x < 3| f(1) = -2| f is decreasing|
x > 3| f(10) = 20| f is increasing|
By the First Derivative Test, x = -3 and x = 3 are local minima and x = 0 is a local maximum.

7) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

The domain of f is [0, 4] whereas the domain of f' is (0, 4). Hence, 0, 2 and 4 are the critical numbers of f. Testing:
0 < x < 2| f'(1) = 0.57...| f is increasing|
2 < x < 4| f'(3) = - 0.57...| f is decreasing|
By the First Derivative Test, x = 2 is a local maximum.

8) We are given the function

First, we find the derivative:

There is no solution for the equation f '(x) = 0.
The domain of f is the collection of all real numbers whereas the domain of f' does not include - 2; therefore, - 2 is the only critical number of f.

Testing:
x < - 2| f'(- 3) = -1| f is decreasing|
x > - 2| f'(0) = 1| f is increasing|
By the First Derivative Test, x = - 2 is a local minimum.

9) We are given the function

First, we find the derivative:

We set the derivative equal to 0 and solve:

Since the domain of f is the same as the domain of f', 0 is the only critical number of f. Testing:
x < 0| f'(-1) = -1| f is decreasing|
x > 0| f'(1) = 1| f is increasing|
By the First Derivative Test, x = 0 is a local minimum.

10) We are given the function

First, we find the derivative:

We set...
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