Question 1

Given f(x) =   find the value of the constant real number k such that f is continuous at x = k.

Accepted Answer

A)  - 1 
B)  -3 
C)  3 
D)  -2 
E)  1 
A)  - 1 
B)  -3 
C)  3 
D)  -2 
E)  1

Question 2

Find the following limit: .

Accepted Answer

A)  -   
B)    
C)    
D)  -   
E)  does not exist 
A)  -   
B)    
C)    
D)  -   
E)  does not exist

Question 3

Let f(x) have values   
Where is f discontinuous?

Accepted Answer

A)  nowhere 
B)  at x = -5 only 
C)  at x = 5 only 
D)  at both x = -5 and x = 5 
E)  none of the above 
A)  nowhere 
B)  at x = -5 only 
C)  at x = 5 only 
D)  at both x = -5 and x = 5 
E)  none of the above

Question 4

Use the formal definition of the limit to verify the addition property of limits: If   f(x) = L and   g(x) = K, then   [f(x) + g(x)] = L + K.

Accepted Answer

The answer of Use the formal definition of the limit...

Question 5

The temperature of an object at time t minutes is   degrees. What is the rate of change of the temperature over the time interval [t, t + h]? How fast is the temperature changing at t = 1?

Accepted Answer

A)  10   degrees/minute, increasing at 5 degrees/minute 
B)  -10   degrees/minute, decreasing at 5 degrees/minute 
C)    degrees/minute, increasing at 5 degrees/minute 
D)  -10   degrees/minute, decreasing at 5 degrees/minute 
E)  10   degrees/minute, decreasing at 5 degrees/minute 
A)  10   degrees/minute, increasing at 5 degrees/minute 
B)  -10   degrees/minute, decreasing at 5 degrees/minute 
C)    degrees/minute, increasing at 5 degrees/minute 
D)  -10   degrees/minute, decreasing at 5 degrees/minute 
E)  10   degrees/minute, decreasing at 5 degrees/minute

Question 6

Find all values of the real number k so that =   .

Accepted Answer

A)  0 
B)  -   
C)  ±   
D)  ± 3 
E)  -3 
A)  0 
B)  -   
C)  ±   
D)  ± 3 
E)  -3

Question 7

Evaluate .

Accepted Answer

A)    
B)    
C)    
D)  0 
E)  does not exist 
A)    
B)    
C)    
D)  0 
E)  does not exist

Question 8

Evaluate   (3   -   + x - 7).

Accepted Answer

A)  -$\infty$ 
B)  0 
C)  1 
D)  $\infty$ 
E)  none of the above 
A)  -$\infty$ 
B)  0 
C)  1 
D)  $\infty$ 
E)  none of the above

Question 9

If   f(x) = $\infty$, then   f(x) does not exist.

Accepted Answer

A) True 
 B)False

Question 10

If   f(x) = 0, then =    =$\infty$

Accepted Answer

A) True 
 B)False

Question 11

Evaluate .

Accepted Answer

A)  -4 
B)  2   
C)  4 
D)  -2   
E)  does not exist 
A)  -4 
B)  2   
C)  4 
D)  -2   
E)  does not exist

Question 12

Use the formal definition of limit to verify that   (x-a) = 0.

Accepted Answer

The answer of Use the formal definition of limit to...

Question 13

Let f and g be continuous functions on the interval ( -$\infty$ , $\infty$ ) such that f has a zero on the closed interval I, and g has a zero on the closed interval.Which of the following functions must have at least one zero on the closed interval I ?

Accepted Answer

A)  f   g , g   f 
B)  f + g , f - g 
C)  f g , g f 
D)    ,   
E)  none of the above. 
A)  f   g , g   f 
B)  f + g , f - g 
C)  f g , g f 
D)    ,   
E)  none of the above.

Question 14

Evaluate .

Accepted Answer

A)  1 
B)  0 
C)  -1 
D)  -2 
E)  does not exist 
A)  1 
B)  0 
C)  -1 
D)  -2 
E)  does not exist

Question 15

Compute the following limit:.

Accepted Answer

A)    a 
B)    a 
C)  -   a 
D)    
E)  does not exist 
A)    a 
B)    a 
C)  -   a 
D)    
E)  does not exist

Question 16

Accepted Answer

A)  only I 
B)  only II 
C)  only I and II 
D)  only II and III 
E)  I, II, and III 
A)  only I 
B)  only II 
C)  only I and II 
D)  only II and III 
E)  I, II, and III

Question 17

Evaluate .

Accepted Answer

A)  -2 
B)    
C)  0 
D)  -   
E)  does not exist 
A)  -2 
B)    
C)  0 
D)  -   
E)  does not exist

Question 18

In which of the intervals (-2, -1), (-1, 0), (0, 1), and (1, 2) does the intermediate value theorem imply that f(x) = 2   - 4   + 5x - 4 must have a zero?

Accepted Answer

A)  (1, 2) 
B)  (0, 1) 
C)  (-1, 0) 
D)  (-2, -1) 
E)  none of the above 
A)  (1, 2) 
B)  (0, 1) 
C)  (-1, 0) 
D)  (-2, -1) 
E)  none of the above

Question 19

Evaluate   (   - x).

Accepted Answer

A)    
B)    
C)  -   
D)  -   
E)  does not exist 
A)    
B)    
C)  -   
D)  -   
E)  does not exist

Question 20

If f(x) is a function defined on the closed interval [a , b] such that f(a)  0 , then the function f must have at least one zero in the interval [a , b].

Accepted Answer

A) True 
 B)False

Given f(x) = find the value of the constant real number k such that f is continuous at x = k.

Find the following limit: .

Let f(x) have values Where is f discontinuous?

Use the formal definition of the limit to verify the addition property of limits: If f(x) = L and g(x) = K, then [f(x) + g(x)] = L + K.

The temperature of an object at time t minutes is degrees. What is the rate of change of the temperature over the time interval [t, t + h]? How fast is the temperature changing at t = 1?

Find all values of the real number k so that = .

Evaluate .

Evaluate (3 - + x - 7).

If $If f(x) = \infty , then f(x) does not exist.$ f(x) = $\infty$ , then $If f(x) = \infty , then f(x) does not exist.$ f(x) does not exist.

If $If f(x) = 0, then = = \infty$ f(x) = 0, then = $If f(x) = 0, then = = \infty$ $If f(x) = 0, then = = \infty$ = $\infty$

Evaluate .

Use the formal definition of limit to verify that (x-a) = 0.

Let f and g be continuous functions on the interval ( - $\infty$ , $\infty$ ) such that f has a zero on the closed interval I, and g has a zero on the closed interval.Which of the following functions must have at least one zero on the closed interval I ?

Evaluate .

Compute the following limit:.

Evaluate .

In which of the intervals (-2, -1), (-1, 0), (0, 1), and (1, 2) does the intermediate value theorem imply that f(x) = 2 - 4 + 5x - 4 must have a zero?

Evaluate ( - x).

If f(x) is a function defined on the closed interval [a , b] such that f(a) < 0 and f(b) > 0 , then the function f must have at least one zero in the interval [a , b].

Preliminaries

Differentiation

Transcendental Functions

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Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

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Multiple Integration

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Exam 2: Limits and Continuity

Given f(x) = find the value of the constant real number k such that f is continuous at x = k.

Find the following limit: .

Let f(x) have values Where is f discontinuous?

Use the formal definition of the limit to verify the addition property of limits: If f(x) = L and g(x) = K, then [f(x) + g(x)] = L + K.

The temperature of an object at time t minutes is degrees. What is the rate of change of the temperature over the time interval [t, t + h]? How fast is the temperature changing at t = 1?

Find all values of the real number k so that = .

Evaluate .

Evaluate (3 - + x - 7).

If f(x) = ∞\infty∞ , then f(x) does not exist.

If f(x) = 0, then = = ∞\infty∞

Evaluate .

Use the formal definition of limit to verify that (x-a) = 0.

Let f and g be continuous functions on the interval ( - ∞\infty∞ , ∞\infty∞ ) such that f has a zero on the closed interval I, and g has a zero on the closed interval.Which of the following functions must have at least one zero on the closed interval I ?

Evaluate .

Compute the following limit:.

Evaluate .

In which of the intervals (-2, -1), (-1, 0), (0, 1), and (1, 2) does the intermediate value theorem imply that f(x) = 2 - 4 + 5x - 4 must have a zero?

Evaluate ( - x).

If f(x) is a function defined on the closed interval [a , b] such that f(a) < 0 and f(b) > 0 , then the function f must have at least one zero in the interval [a , b].

Preliminaries

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

If $If f(x) = \infty , then f(x) does not exist.$ f(x) = $\infty$ , then $If f(x) = \infty , then f(x) does not exist.$ f(x) does not exist.

If $If f(x) = 0, then = = \infty$ f(x) = 0, then = $If f(x) = 0, then = = \infty$ $If f(x) = 0, then = = \infty$ = $\infty$

Let f and g be continuous functions on the interval ( - $\infty$ , $\infty$ ) such that f has a zero on the closed interval I, and g has a zero on the closed interval.Which of the following functions must have at least one zero on the closed interval I ?