Calculs Review Jeopardy

Theorems

1

f(b) - f(a) / b - a

Average Rate of Change

2

What does the following notation mean?

Lim   f(x) = 5

x->2               

As x approaches 2, f(x) approaches 5

3

Write the notation for the following limit. 

The limit of 𝑓 as 𝑥 approaches 3 from the left side is 10.

Lim f(x) = 10

x->3(-)           

4

List a non removable discontinuity

vertical asymptote or jump discontinuity

5

Limit as x approaches infinity refers to ......

Horizontal Asymptote 

Solving

1

Lim  (x²+2x-4)

x->-1              

-5

2

f(x) = 1)  1^x          ,      x < 4  or x = 4      

           2) 17 - x²    ,      x > 4    

Continuous at x = 4? Why or why not?                             

yes, f(4) = 1    and 

lim  f(x)  = f(4)

x->4                 

3

g(x) = 1)    -|x|     ,      x < 5 or x = 5       

           2) 20 - x²    ,    -5 < x < 3 or x = 3

           3) 4x - 1     ,     x > 3            

Find g(3).                                

20 - 3² = 11

4

Identify the Vertical asymptote of the given function.

f(x) = ( x³+2x²-24x )  /  ( x²-x )

Vertical asymptote at x=1

5

Evaluate the follwing limit.

Lim       ( 6x²+13x-5 ) / ( 3x-1)

x-> 1/3

17 / 3

Continuity

1

List a type of removable discontinuity

a hole

2

What value must k be so that f(x) is continuous at x = -2.

Lim  (x²-4) / (x+2) 

x->-2                    

k = -4

3

What is the value of the lim(x→a) (e^x-1)/x

1

4

This type of discontinuity often seen in the piecewise functions, where the left-hand limit and the right-hand limit at x=c exist, but they are not equal to each other.

Jump Discontinuity

5

f(x) =  1)     3-x²      ,    x ≤ 4 

           2)     x+k      ,    x > 4

Find the value of k that makes this function continuous.

k = -17

Limits

1

Lim f(x)      exists when .....

x->a                             

left side = right side

2

If f(x) = 3x²-10x+2  , [a,b] = [-1,3],    k=1

Determine if the intermediate value theorem holds for the given value of k. 

Either IVT applies because k exists between [ , ]

or  IVT does not apply because k doesnt exist between [ , ]

IVT applies because k exists between (-1,15)

3

Evaluate the following limit

Lim       (x-2)  /  (x²-3x+2)

x->1(+)                            

Infinity

4

Identify the horizontal asymptotes in the following function.

f(x) = (2x+5)(2-6x)  /  (3x-2)²

y = -4/3

5

Identify the type of discontinuity and where it is located.

f(x) = 1)  x² - 8x - 10          ,      x < -1       

           2) - x² -6x - 6  ,          x > -1     

Hole at x = -1

Differentiation

1

Find the average rate of change on the given interval.

A(t) = 2^t ;  [2,4]

t represents years

A represents dollars

6 dollars per year

2

Estimate the derivate at a given point.

f(x) =  8x-3 ;  f'(2)

8

3

Estimate the derivate at a given point.

f(x) =  Ln(√x)  ;   Find f'(1).

0.5 or 1/2

4

Find the instantaneous rate of change of each function at the given x-value. 

f(x) =  x² - x   at x =-1

-3

5

Find the derivative of the following function.

y = 5x² - x

dy/dx = 10x - 1