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Lesson: Finding Functions given Their Derivatives

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

Find the function of the curve whose first derivative is and the function equals 7 when equals .

  • A
  • B
  • C
  • D

Q2:

Determine the function 𝑓 if 𝑓 β€² ( π‘₯ ) = βˆ’ 3 π‘₯ + 1 √ π‘₯ and 𝑓 ( 1 ) = 4 .

  • A 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + 2 √ π‘₯ + 4 3 2
  • B 𝑓 ( π‘₯ ) ) = βˆ’ 2 π‘₯ + 2 √ π‘₯ βˆ’ 4 3 2
  • C 𝑓 ( π‘₯ ) = βˆ’ 9 π‘₯ 2 + 2 √ π‘₯ + 4 3 2
  • D 𝑓 ( π‘₯ ) = βˆ’ 9 π‘₯ 2 + 2 √ π‘₯ + 1 3 3 2
  • E 𝑓 ( π‘₯ ) = βˆ’ 2 π‘₯ + 2 √ π‘₯ + 1 3 3 2

Q3:

Determine the function 𝑓 if 𝑓 β€² β€² ( π‘₯ ) = βˆ’ π‘₯ + 1 4 , 𝑓 ( 1 ) = 5 , and 𝑓 β€² ( 1 ) = 3 .

  • A 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 3 0 + π‘₯ 2 + 1 1 π‘₯ 5 + 7 3 6 2
  • B 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 3 0 + π‘₯ 2 βˆ’ 1 1 π‘₯ 5 βˆ’ 7 3 6 2
  • C 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 5 + 1 6 π‘₯ 5 5
  • D 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 5 + 2 6 π‘₯ 5 + 1 3 5
  • E 𝑓 ( π‘₯ ) = βˆ’ π‘₯ 3 0 + π‘₯ 2 + 2 1 π‘₯ 5 + 1 3 6 2

Q4:

Determine the function 𝑓 satisfying 𝑓 β€² β€² ( π‘₯ ) = 1 2 π‘₯ βˆ’ 1 0 π‘₯ + 3 2 , 𝑓 ( 0 ) = 5 , and 𝑓 β€² ( 0 ) = 2 .

  • A 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ 3 + 3 π‘₯ 2 + 2 π‘₯ + 5 4 3 2
  • B 𝑓 ( π‘₯ ) = 1 2 π‘₯ βˆ’ 1 0 π‘₯ + 3 π‘₯ + 2 π‘₯ + 5 4 3 2
  • C 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 5 π‘₯ + 3 π‘₯ + 2 3 2
  • D 𝑓 ( π‘₯ ) = 4 π‘₯ βˆ’ 5 π‘₯ + 3 π‘₯ + 5 3 2
  • E 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 5 π‘₯ 3 + 3 π‘₯ 2 + 5 π‘₯ + 2 4 3 2

Q5:

Find the equation of the curve through the point ( βˆ’ 4 , βˆ’ 4 ) with the property that the slope of the normal to the curve at ( π‘₯ , 𝑦 ) is βˆ’ 6 2 π‘₯ βˆ’ 3 .

  • A 𝑦 = π‘₯ 6 βˆ’ π‘₯ 2 βˆ’ 2 6 3 2
  • B 𝑦 = π‘₯ 3 βˆ’ π‘₯ 2 βˆ’ 3 4 3 2
  • C 𝑦 = 2 π‘₯ 3 βˆ’ π‘₯ 2 βˆ’ 1 6 2
  • D 𝑦 = π‘₯ βˆ’ 3 π‘₯ βˆ’ 3 2 2

Q6:

If the rate of change in the area 𝐴 of a metallic plate with respect to time due to heating is given by the relation d d 𝐴 𝑑 = 0 . 0 3 6 𝑑 + 0 . 0 3 8 𝑑 , 2 where the area 𝐴 is in square metres, and the time 𝑑 is in minutes, given that 𝐴 = 6 7 m 2 when 𝑑 = 8 m i n u t e s , find, correct to the nearest two decimal places, the area of the plate just before heating.

Q7:

A group of labourers are digging a hole, where the rate of change of the volume 𝑉 of the sand removed in cubic metres with respect to the time 𝑑 in hours is given by the relation d d 𝑉 𝑑 = 𝑑 + 1 5 . Calculate the volume of the sand dug out in 5 hours rounded to the nearest hundredth.

Q8:

Suppose that d d s i n c o s 𝑦 π‘₯ = βˆ’ 9 2 π‘₯ βˆ’ 3 5 π‘₯ and 𝑦 = 7 when π‘₯ = πœ‹ 6 . Find 𝑦 in terms of π‘₯ .

  • A 𝑦 = βˆ’ 3 5 5 π‘₯ + 9 2 2 π‘₯ + 1 0 1 2 0 s i n c o s
  • B 𝑦 = 9 2 5 π‘₯ + 3 5 2 π‘₯ + 8 9 2 0 s i n c o s
  • C 𝑦 = βˆ’ 9 2 π‘₯ βˆ’ 3 5 π‘₯ + 1 3 s i n c o s
  • D 𝑦 = βˆ’ 3 5 2 π‘₯ βˆ’ 9 2 5 π‘₯ + 8 9 2 0 s i n c o s
  • E 𝑦 = βˆ’ 9 2 2 π‘₯ βˆ’ 3 5 5 π‘₯ + 8 9 2 0 s i n c o s

Q9:

Determine 𝑓 ( 𝑑 ) if 𝑓 β€² β€² β€² ( 𝑑 ) = βˆ’ 4 √ 𝑑 + 5 𝑑 c o s .

  • A 𝑓 ( 𝑑 ) = βˆ’ 3 2 𝑑 1 0 5 βˆ’ 5 𝑑 + 𝑑 + 𝑑 + 7 2 s i n C D E 2
  • B 𝑓 ( 𝑑 ) = βˆ’ 3 2 𝑑 1 0 5 βˆ’ 5 𝑑 + 𝑑 + 7 2 s i n C D
  • C 𝑓 ( 𝑑 ) = βˆ’ 8 𝑑 3 + 5 𝑑 + 3 2 s i n C
  • D 𝑓 ( 𝑑 ) = βˆ’ 3 2 𝑑 1 0 5 βˆ’ 5 𝑑 + 𝑑 + 7 2 s i n C E 2
  • E 𝑓 ( 𝑑 ) = βˆ’ 1 6 𝑑 1 5 βˆ’ 5 𝑑 + 𝑑 + 5 2 c o s C D

Q10:

Given that d d c o s 𝑠 𝑑 = βˆ’ 9 ο€Ό 9 𝑑 4  , and 𝑠 = βˆ’ 4 when 𝑑 = 4 πœ‹ 3 , find the relation between 𝑠 and 𝑑 .

  • A 𝑠 ( 𝑑 ) = βˆ’ 4 ο€Ό 9 𝑑 4  βˆ’ 4 s i n
  • B 𝑠 ( 𝑑 ) = 9 ο€Ό 9 𝑑 4  βˆ’ 4 s i n
  • C 𝑠 ( 𝑑 ) = 4 ο€Ό 9 𝑑 4  βˆ’ 4 s i n
  • D 𝑠 ( 𝑑 ) = βˆ’ 9 ο€Ό 9 𝑑 4  βˆ’ 4 s i n

Q11:

Determine the function 𝑓 if 𝑓 β€² β€² ( 𝑑 ) = 5 √ 𝑑 βˆ’ 4 𝑑 3 c o s , 𝑓 ( 0 ) = 2 , and 𝑓 ( 1 ) = 3 .

  • A 𝑓 ( 𝑑 ) = 4 5 𝑑 2 8 + 4 𝑑 + ο€Ό 9 5 2 8 βˆ’ 4 1  𝑑 βˆ’ 2 7 3 c o s c o s
  • B 𝑓 ( 𝑑 ) = 3 𝑑 2 8 βˆ’ 𝑑 + ο€Ό 9 5 2 8 βˆ’ 4 1  𝑑 βˆ’ 2 4 3 c o s c o s
  • C 𝑓 ( 𝑑 ) = 1 5 𝑑 4 βˆ’ 4 𝑑 + ο€Ό 9 5 2 8 βˆ’ 4 1  𝑑 βˆ’ 2 4 3 s i n c o s
  • D 𝑓 ( 𝑑 ) = 4 5 𝑑 2 8 βˆ’ 4 𝑑 + ο€Ό 6 7 2 8 βˆ’ 4 1  𝑑 βˆ’ 1 7 3 s i n c o s
  • E 𝑓 ( 𝑑 ) = 4 5 𝑑 2 8 + 4 𝑑 + ο€Ό 6 7 2 8 βˆ’ 4 1  𝑑 βˆ’ 1 7 3 c o s c o s

Q12:

Find the function on which satisfies , , and .

  • A
  • B
  • C
  • D
  • E

Q13:

A function is such that, at all points, the product of the slope of its graph and the square of its π‘₯ -coordinate is 3. Find the equation of the curve given the curve passes through the point ( 1 , βˆ’ 8 ) .

  • A 𝑦 = βˆ’ 3 π‘₯ βˆ’ 5
  • B 𝑦 = βˆ’ 3 π‘₯ βˆ’ 1 1
  • C 𝑦 = 3 π‘₯ βˆ’ 1 1
  • D 𝑦 = 3 π‘₯ βˆ’ 5

Q14:

The rate of change of the volume of a gas, 𝑉 in cubic metres, with respect to its pressure 𝑝 is given by d d 𝑉 𝑝 = βˆ’ 2 π‘Ž 9 𝑝 2 , where π‘Ž is a constant. Find the relationship between the volume of the gas and the pressure when 𝑉 = 1 3 m 3 and 𝑝 = 1 8 / N m 2 .

  • A 𝑉 = βˆ’ 1 6 π‘Ž 9 + 2 π‘Ž 9 𝑝 + 1 3
  • B 𝑉 = 1 6 π‘Ž 9 + 2 π‘Ž 9 𝑝 + 1 3
  • C 𝑉 = βˆ’ 1 6 π‘Ž 9 βˆ’ 2 π‘Ž 9 𝑝 + 1 3
  • D 𝑉 = 1 6 π‘Ž 9 βˆ’ 2 π‘Ž 9 𝑝 + 1 3

Q15:

Find the equation of the curve passing though the point ( 0 , 1 ) given the gradient of the tangent at any point is 2 π‘₯ βˆ’ 3 . Then find the tangents at the points on the curve which intersect with the line 𝑦 = 5 .

  • Athe curve: 𝑦 = π‘₯ βˆ’ 3 π‘₯ + 1 2 , the tangents: 𝑦 βˆ’ 5 π‘₯ + 1 5 = 0 and 𝑦 + 5 π‘₯ = 0
  • Bthe curve: 𝑦 = π‘₯ βˆ’ 3 π‘₯ + 1 2 , the tangents: 𝑦 + 5 π‘₯ βˆ’ 2 5 = 0 and 𝑦 βˆ’ 5 π‘₯ βˆ’ 1 0 = 0
  • Cthe curve: 𝑦 = 2 π‘₯ βˆ’ 3 π‘₯ βˆ’ 1 2 , the tangents: βˆ’ 5 𝑦 βˆ’ π‘₯ + 2 9 = 0 and 5 𝑦 βˆ’ π‘₯ βˆ’ 2 6 = 0
  • Dthe curve: 𝑦 = π‘₯ βˆ’ 3 π‘₯ + 1 2 , the tangents: 5 𝑦 βˆ’ π‘₯ βˆ’ 2 1 = 0 and βˆ’ 5 𝑦 βˆ’ π‘₯ + 2 4 = 0

Q16:

The slope at each point ( π‘₯ , 𝑦 ) of a curve is inversely proportional to π‘₯ . The slope is 8 when π‘₯ = 7 and 𝑦 = βˆ’ 1 . Determine 𝑦 in terms of π‘₯ .

  • A 5 6 | π‘₯ | βˆ’ 5 6 7 βˆ’ 1 l n l n
  • B 5 6 | π‘₯ | βˆ’ 1 + 5 6 7 l n l n
  • C l n l n | π‘₯ | βˆ’ 7 βˆ’ 1
  • D 5 6 | π‘₯ | l n

Q17:

Determine the function 𝑓 , if 𝑓 β€² β€² ( 𝑑 ) = βˆ’ 3 𝑑 + 1 𝑑 2 2 , when 𝑑 > 0 , 𝑓 ( 2 ) = 2 , and 𝑓 β€² ( 1 ) = 2 .

  • A 𝑓 ( 𝑑 ) = βˆ’ 𝑑 4 + 4 𝑑 βˆ’ 𝑑 βˆ’ 2 + 2 4 l n l n
  • B 𝑓 ( 𝑑 ) = 𝑑 1 2 + 4 𝑑 βˆ’ 𝑑 βˆ’ 2 + 2 4 l n l n
  • C 𝑓 ( 𝑑 ) = βˆ’ 𝑑 + 4 βˆ’ 1 𝑑 3
  • D 𝑓 ( 𝑑 ) = βˆ’ 𝑑 + 1 9 2 βˆ’ 1 𝑑 3
  • E 𝑓 ( 𝑑 ) = βˆ’ 𝑑 4 + 1 9 𝑑 2 βˆ’ 𝑑 βˆ’ 1 3 + 2 4 l n l n

Q18:

The function 𝑦 = 𝑓 ( π‘₯ ) satisfies d d 2 2 𝑦 π‘₯ = π‘Ž π‘₯ + 𝑏 for some constants π‘Ž and 𝑏 . If the graph of 𝑓 has an inflection at ( 0 , 9 ) and a local minimum value at ( 3 , βˆ’ 9 ) , what is 𝑓 ( π‘₯ ) ? If the graph also has a local maximum, identify it.

  • A 𝑓 ( π‘₯ ) = 1 3 π‘₯ βˆ’ 9 π‘₯ + 9 3 , local maximum value: 𝑓 ( βˆ’ 3 ) = 2 7
  • B 𝑓 ( π‘₯ ) = π‘₯ + 1 8 π‘₯ + 9 3 , local maximum value: 𝑓 ( βˆ’ 3 ) = βˆ’ 5 4
  • C 𝑓 ( π‘₯ ) = 2 3 π‘₯ βˆ’ 3 6 π‘₯ + 9 3 , local maximum value: 𝑓 ( βˆ’ 3 ) = 1 0 8
  • D 𝑓 ( π‘₯ ) = 1 2 π‘₯ βˆ’ 1 8 π‘₯ + 9 3 , local maximum value: 𝑓 ( βˆ’ 3 ) = 5 4

Q19:

The equation of the tangent to a curve at ( βˆ’ 2 , 4 ) is 𝑦 = βˆ’ 5 π‘₯ βˆ’ 6 . Find the equation of the curve given 𝑓 β€² β€² ( π‘₯ ) = 9 π‘₯ .

  • A 𝑦 = 3 π‘₯ 2 βˆ’ 2 3 π‘₯ βˆ’ 3 0 3
  • B 𝑦 = 9 π‘₯ 2 βˆ’ 2 3 π‘₯ βˆ’ 6 3
  • C 𝑦 = 3 π‘₯ 2 βˆ’ 2 3 π‘₯ βˆ’ 3 4 3
  • D 𝑦 = 3 π‘₯ βˆ’ 2 3 π‘₯ βˆ’ 1 8 3

Q20:

The rate of change of the gradient to the curve that passes through the points ( 2 , βˆ’ 2 ) and ( βˆ’ 1 , βˆ’ 5 ) is 1 2 π‘₯ βˆ’ 1 2 . Find the equation of the curve.

  • A 𝑦 = 2 π‘₯ βˆ’ 6 π‘₯ + π‘₯ + 4 3 2
  • B 𝑦 = 4 π‘₯ βˆ’ 6 π‘₯ βˆ’ 4 π‘₯ βˆ’ 1 3 2
  • C 𝑦 = 4 π‘₯ βˆ’ 6 π‘₯ + 1 2
  • D 𝑦 = 2 π‘₯ βˆ’ 6 π‘₯ + 4 2
  • E 𝑦 = 1 2 π‘₯ βˆ’ 1 2 π‘₯ + π‘₯ + 4 3 2

Q21:

The graph of 𝑓 passes through ( 2 , 5 ) and the slope of the tangent at ( π‘₯ , 𝑓 ( π‘₯ ) ) is π‘₯ βˆ’ 1 . What is 𝑓 ( 1 ) ?

  • A 9 2
  • B 5 2
  • C βˆ’ 6
  • D19
  • E 7 2

Q22:

The function 𝑓 ( π‘₯ ) satisfies the relation 𝑓 ( π‘Ž + β„Ž ) βˆ’ 𝑓 ( π‘Ž ) = βˆ’ 2 π‘Ž β„Ž π‘˜ + 2 β„Ž 2 where π‘Ž , β„Ž ∈ ℝ and π‘˜ is a constant. Find 𝑓 ( π‘₯ ) given 𝑓 ( βˆ’ 4 ) = 8 and 𝑓 ( 2 ) = βˆ’ 4 .

  • A π‘₯ βˆ’ 8 2
  • B βˆ’ π‘₯
  • C βˆ’ 2 π‘₯
  • D βˆ’ π‘₯ + 1 2 2

Q23:

Determine the function 𝑓 ( π‘₯ ) such that 𝑓 ( π‘₯ ) = 1 5 π‘₯ βˆ’ 2 π‘₯ βˆ’ 3 β€² β€² 2 , 𝑓 ( 0 ) = 5 , and 𝑓 ( 1 ) = 3 .

  • A 𝑓 ( π‘₯ ) = 5 π‘₯ 4 βˆ’ π‘₯ 3 βˆ’ 3 π‘₯ 2 βˆ’ 1 7 π‘₯ 1 2 + 5 4 3 2
  • B 𝑓 ( π‘₯ ) = 5 π‘₯ 4 βˆ’ 2 π‘₯ 3 βˆ’ 3 π‘₯ 2 + 7 9 π‘₯ 1 2 + 5 4 3 2
  • C 𝑓 ( π‘₯ ) = 5 π‘₯ βˆ’ π‘₯ βˆ’ 1 7 π‘₯ βˆ’ 1 7 1 2 3 2
  • D 𝑓 ( π‘₯ ) = 5 π‘₯ 4 βˆ’ π‘₯ 6 βˆ’ 3 π‘₯ 2 + 7 1 2 4 3 2
  • E 𝑓 ( π‘₯ ) = 5 π‘₯ 4 βˆ’ π‘₯ 3 βˆ’ 3 π‘₯ 2 + 7 π‘₯ 1 2 + 3 4 3 2

Q24:

The second derivative of a function is 6 π‘₯ and the equation of the tangent to its graph at ( βˆ’ 2 , βˆ’ 4 ) is 6 π‘₯ βˆ’ 𝑦 + 8 = 0 . Find the equation of the curve.

  • A 𝑦 = π‘₯ βˆ’ 6 π‘₯ βˆ’ 8 
  • B 𝑦 = π‘₯ βˆ’ 8 
  • C 𝑦 = π‘₯ βˆ’ 8 π‘₯ βˆ’ 6 
  • D 𝑦 = π‘₯ + 1 8 π‘₯ + 4 0 
  • E 𝑦 = π‘₯ βˆ’ 1 2 π‘₯ + 4 

Q25:

Determine the function 𝑓 ( 𝑑 ) such that 𝑓 β€² ( 𝑑 ) = βˆ’ 2 3 ( 𝑑 + 1 ) 2 , and 𝑓 ( 1 ) = 0 .

  • A 𝑓 ( 𝑑 ) = βˆ’ 2 𝑑 3 + πœ‹ 6 t a n βˆ’ 1
  • B 𝑓 ( 𝑑 ) = βˆ’ 2 𝑑 3 βˆ’ πœ‹ 3 s i n βˆ’ 1
  • C 𝑓 ( 𝑑 ) = βˆ’ 2 𝑑 3 + πœ‹ 3 s i n βˆ’ 1
  • D 𝑓 ( 𝑑 ) = βˆ’ 2 𝑑 3 βˆ’ πœ‹ 6 t a n βˆ’ 1
  • E 𝑓 ( 𝑑 ) = βˆ’ 2 𝑑 3 + 1 s i n βˆ’ 1
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