ESTIMATE THE DERIVATIVE OF A FUNCTION AT A GIVEN POINT FROM TABLE

To estimate the derivative of a function at a given point from table, we have to follow the steps given below.

Step 1 :

Determine the slope between point and the point directly to the left of point .

Step 2 :

Determine the slope between point and the point directly to the right of point.

Average rate of change =f(b)-f(a)b-a(or)Average rate of change =yx (or)Average rate of change =y2-y1x2-x1

Use the tables to estimate the value of the derivative at the given point. Indicate units of measures.

Problem 1 :

estimate-the-deri-of-a-fun-q1

a.   f'(8)

b.   f'(3.5)

Solution :

a. To find f'(8) :

Here 8 lies between 7 and 9.

(7, 807) and (9, 902)

f' (x) = f(b) - f(a)b - af'(8)= f(9) - f(7)9 - 7= 902 - 8072=95 2= 47.5

f'(8) = 47.5 visitors/hours

b. To find f'(3.5) :

f' (x) = f(b) - f(a)b - af'(3.5)= f(4) - f(3)4 - 3= 595 - 4761=595 - 476= 119

f'(3.5) = 119 visitors/hours

Problem 2 :

estimate-the-deri-of-a-fun-q2

a. f'(17) 

b. f'(24.5) 

Solution :

a. To find f'(17) :

f' (x) = f(b) - f(a)b - af'(17)= f(23) - f(11)23 - 11= 51 - 7112=-2012= -1.667

f'(17) = -1.667  ºC per cm

b. To find f'(24.5) :

f' (x) = f(b) - f(a)b - af'(24.5)= f(26) - f(23)26 - 23= 40 - 513=-113= -3.667

f'(24.5) = -3.667  ºC per cm

Problem 3 :

estimate-the-deri-of-a-fun-q3

a. f'(1.5) 

b. f'(11) 

Solution :

a. To find f'(1.5) :

f' (x) = f(b) - f(a)b - af'(1.5)= f(3) - f(0)3 - 0= 20 - 53=153= 5

f'(1.5) = 5 students/year2

b. To find f'(11) :

f' (x) = f(b) - f(a)b - af'(11)= f(15) - f(7)15 - 7= -2 - 78=-98= -1.125

f'(11) = -1.125 students/year2

Problem 4 :

estimate-the-deri-of-a-fun-q4

a. f'(47.5) 

b.  f'(9) 

Solution :

a. To find f'(47.5) :

f' (x) = f(b) - f(a)b - af'(47.5)= f(50) - f(45)50 - 45= 36 - 215=155= 3

f'(47.5) = 3 pages per day

b. To find f'(9) :

f' (x) = f(b) - f(a)b - af'(9)= f(13) - f(5)13 - 5= 20 - 518=-318= -3.875

f'(9) = -3.875 pages per day

Problem 5 :

estimate-the-deri-of-a-fun-q5

a. f'(20) 

b. f'(82.5) 

Solution :

a. To find f'(20) :

f' (x) = f(b) - f(a)b - af'(20)= f(30) - f(10)30 - 10= 790 - 100520=-21520= -10.75

f'(20) = -10.75 gallons/sec2

b. To find f'(82.5) :

f' (x) = f(b) - f(a)b - af'(82.5)= f(100) - f(65)100 - 65= 209 - 43435=-22535= -6.4285

f'(82.5) = -6.4285 gallons/sec2

Problem 6 :

estimate-the-deri-of-a-fun-q6

a. f'(25.5) 

b. f'(13.5) 

Solution :

a. To find f'(25.5) :

f' (x) = f(b) - f(a)b - af'(25.5)= f(30) - f(21)30 - 21= 272 - 1509=1229= 13.555

f'(25.5) = 13.555 yards per carry

b. To find f'(13.5) :

f' (x) = f(b) - f(a)b - af'(13.5)= f(15) - f(12)15 - 12= 98 - 1073=-93= -3

f'(13.5) = -3 yards per carry

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