Existence Theorem Assessment Test

1) The graph of f is concave upward or downward on I if f ' is increasing on the interval...

Concave upwards

Convex upwards

Concave downwards

Convex downwards

The graph of a function is concave upward on an interval if the derivative of the function is increasing on that interval. This means that as we move from left to right on the interval, the slope of the tangent lines to the graph of the function is increasing. Therefore, the correct answer is "Concave upwards".

Explanation

The graph of a function is concave upward on an interval if the derivative of the function is increasing on that interval. This means that as we move from left to right on the interval, the slope of the tangent lines to the graph of the function is increasing. Therefore, the correct answer is "Concave upwards".

2) Find the differential dy of the given function. y= 3x 2 - 4

40x dx

12x dx

6x dx

36x dx

The given function is y = 3x^2 - 4. To find the differential dy, we differentiate the function with respect to x. The derivative of 3x^2 is 6x, and the derivative of -4 is 0. Therefore, dy = 6x dx.

Explanation

The given function is y = 3x^2 - 4. To find the differential dy, we differentiate the function with respect to x. The derivative of 3x^2 is 6x, and the derivative of -4 is 0. Therefore, dy = 6x dx.

3) What is the average rate of change of the function below over the indicated interval? f(x) = 12√(x), [1,16]?

12/5

0

1

24

The average rate of change of a function over an interval is found by taking the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values. In this case, the function is f(x) = 12√(x) and the interval is [1,16]. Evaluating the function at the endpoints, we get f(1) = 12√(1) = 12 and f(16) = 12√(16) = 48. The difference in the function values is 48 - 12 = 36 and the difference in the input values is 16 - 1 = 15. Dividing the difference in function values by the difference in input values, we get 36/15 = 12/5. Therefore, the average rate of change of the function over the interval [1,16] is 12/5.

Explanation

The average rate of change of a function over an interval is found by taking the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values. In this case, the function is f(x) = 12√(x) and the interval is [1,16]. Evaluating the function at the endpoints, we get f(1) = 12√(1) = 12 and f(16) = 12√(16) = 48. The difference in the function values is 48 - 12 = 36 and the difference in the input values is 16 - 1 = 15. Dividing the difference in function values by the difference in input values, we get 36/15 = 12/5. Therefore, the average rate of change of the function over the interval [1,16] is 12/5.

4) Find the limit. lim(4 + 3/x) x->∞

1

2

3

4

As x approaches infinity, the term 3/x approaches 0. Therefore, the limit of (4 + 3/x) as x approaches infinity is equal to 4. This is because the constant term 4 does not change as x gets larger and larger, and the term 3/x becomes negligible.

Explanation

As x approaches infinity, the term 3/x approaches 0. Therefore, the limit of (4 + 3/x) as x approaches infinity is equal to 4. This is because the constant term 4 does not change as x gets larger and larger, and the term 3/x becomes negligible.

5) Find any critical numbers of the function. f(x)= x 3 - 3x 2

X=1 , x=2

X=0 , x =3

X=1 , x=3

X=0 , x=2

The critical numbers of a function are the values of x where the derivative is either zero or undefined. To find the critical numbers of the function f(x) = x^3 - 3x^2, we need to find the derivative of the function and set it equal to zero. Taking the derivative of f(x), we get f'(x) = 3x^2 - 6x. Setting this equal to zero, we can solve for x and find that x = 0 and x = 2. Therefore, the critical numbers of the function are x = 0 and x = 2.

Explanation

The critical numbers of a function are the values of x where the derivative is either zero or undefined. To find the critical numbers of the function f(x) = x^3 - 3x^2, we need to find the derivative of the function and set it equal to zero. Taking the derivative of f(x), we get f'(x) = 3x^2 - 6x. Setting this equal to zero, we can solve for x and find that x = 0 and x = 2. Therefore, the critical numbers of the function are x = 0 and x = 2.

6) What is x-value at which the function below has the same slope as the average rate of change over the indicated interval? f(x) = 12√(x), [1,16]?

24

18.78

6.25

7.35

The average rate of change over the interval [1,16] can be found by taking the difference in the y-values divided by the difference in the x-values. In this case, the average rate of change is (f(16) - f(1))/(16-1) = (12√(16) - 12√(1))/(16-1) = (12*4 - 12*1)/15 = 48/15 = 3.2. The slope of the function f(x) = 12√(x) can be found by taking the derivative, which is f'(x) = 6/√(x). Setting this equal to the average rate of change, we get 6/√(x) = 3.2. Solving for x, we find x = 6.25.

Explanation

The average rate of change over the interval [1,16] can be found by taking the difference in the y-values divided by the difference in the x-values. In this case, the average rate of change is (f(16) - f(1))/(16-1) = (12√(16) - 12√(1))/(16-1) = (12*4 - 12*1)/15 = 48/15 = 3.2. The slope of the function f(x) = 12√(x) can be found by taking the derivative, which is f'(x) = 6/√(x). Setting this equal to the average rate of change, we get 6/√(x) = 3.2. Solving for x, we find x = 6.25.

7) If there is an open interval containing c on which f(c) is a maximun, then f(c) is called...

Absolute minimum

Relative maximum

Absolute maximum

Relative minimum

If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum. This means that f(c) is the highest value within that interval, but it may not be the highest value in the entire function. An absolute maximum would be the highest value in the entire function.

Explanation

If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum. This means that f(c) is the highest value within that interval, but it may not be the highest value in the entire function. An absolute maximum would be the highest value in the entire function.

8) If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.

The mean value theorem

Pythagoraean theorem

Binomial theorem

The extreme value theorem

The extreme value theorem states that if a function is continuous on a closed interval, then it must have both a minimum and a maximum value on that interval. Therefore, the given statement aligns with the extreme value theorem.

Explanation

The extreme value theorem states that if a function is continuous on a closed interval, then it must have both a minimum and a maximum value on that interval. Therefore, the given statement aligns with the extreme value theorem.

9) What is the average rate of change of the function below over the indicated interval? f(x) = x2 - 3x - 28, [-4,7]

11

66

6

0

The average rate of change of a function over an interval is determined by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values. In this case, the function is f(x) = x^2 - 3x - 28 and the interval is [-4,7]. Plugging in the endpoints into the function, we get f(-4) = 12 and f(7) = 12. The difference in the function values is 12 - 12 = 0. The difference in the input values is 7 - (-4) = 11. Dividing the difference in function values by the difference in input values, we get 0/11 = 0. Therefore, the average rate of change of the function over the interval is 0.

Explanation

The average rate of change of a function over an interval is determined by finding the difference in the function values at the endpoints of the interval and dividing it by the difference in the input values. In this case, the function is f(x) = x^2 - 3x - 28 and the interval is [-4,7]. Plugging in the endpoints into the function, we get f(-4) = 12 and f(7) = 12. The difference in the function values is 12 - 12 = 0. The difference in the input values is 7 - (-4) = 11. Dividing the difference in function values by the difference in input values, we get 0/11 = 0. Therefore, the average rate of change of the function over the interval is 0.

10) The Rolle's Theorem is named after who?

Michel Rolle

Janet Rolle

Albert Rolle

Pierce Rolle

The correct answer is Michel Rolle. Michel Rolle was a French mathematician who lived in the 17th century. He is known for his work in algebra and calculus, and his contributions to mathematics include the development of Rolle's Theorem. Rolle's Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, and the function values at the endpoints of the interval are equal, then there exists at least one point in the interval where the derivative of the function is zero.

Explanation

The correct answer is Michel Rolle. Michel Rolle was a French mathematician who lived in the 17th century. He is known for his work in algebra and calculus, and his contributions to mathematics include the development of Rolle's Theorem. Rolle's Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, and the function values at the endpoints of the interval are equal, then there exists at least one point in the interval where the derivative of the function is zero.