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Lesson: Applications of the Indefinite Integration

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

A curve passes through ( 0 , 1 ) and the tangent at its point ( π‘₯ , 𝑦 ) has slope 6 π‘₯ √ 8 π‘₯ + 1 2 . What is the equation of the curve?

  • A 𝑦 = 1 4 ο€Ή 8 π‘₯ + 1  + 3 4 2 3 2
  • B 𝑦 = 1 3 2 ο€Ή 8 π‘₯ + 1  + 3 1 3 2 2 3 2
  • C 𝑦 = 3 1 6 ο€Ή 8 π‘₯ + 1  + 1 3 1 6 2 3 2
  • D 𝑦 = 1 4 ο€Ή 8 π‘₯ + 1  + 5 4 2 3 2

Q2:

A curve passes through ( 0 , 1 ) and the tangent at its point ( π‘₯ , 𝑦 ) has slope 4 π‘₯ √ 2 π‘₯ + 9 2 . What is the equation of the curve?

  • A 𝑦 = 2 3 ο€Ή 2 π‘₯ + 9  βˆ’ 1 7 2 3 2
  • B 𝑦 = 1 3 ο€Ή 2 π‘₯ + 9  βˆ’ 8 2 3 2
  • C 𝑦 = 2 ο€Ή 2 π‘₯ + 9  βˆ’ 5 3 2 3 2
  • D 𝑦 = 2 3 ο€Ή 2 π‘₯ + 9  + 1 9 2 3 2

Q3:

Find the equation of the curve that passes through the point ( βˆ’ 2 , 1 ) given that the gradient of the tangent to the curve is βˆ’ 1 1 π‘₯ 2 .

  • A 𝑦 = βˆ’ 1 1 3 π‘₯ βˆ’ 8 5 3 3
  • B 𝑦 = βˆ’ 1 1 3 π‘₯ + 4 7 3 2
  • C 𝑦 = βˆ’ 1 1 π‘₯ + 9 3
  • D 𝑦 = βˆ’ 1 1 3 π‘₯ + 3 C

Q4:

The gradient of the tangent to a curve is βˆ’ 6 π‘₯ + 6 π‘₯ s i n c o s . For π‘₯ ∈  0 , πœ‹ 3  , the curve has a local minimum value of βˆ’ 4 6 √ 2 9 . Find the equation of the curve.

  • A 𝑦 = 1 6 6 π‘₯ + 1 6 6 π‘₯ βˆ’ 8 9 √ 2 1 8 s i n c o s
  • B 𝑦 = 1 6 6 π‘₯ + 1 6 6 π‘₯ βˆ’ 9 5 √ 2 1 8 s i n c o s
  • C 𝑦 = βˆ’ 1 6 6 π‘₯ βˆ’ 1 6 6 π‘₯ βˆ’ 8 9 √ 2 1 8 s i n c o s
  • D 𝑦 = βˆ’ 1 6 6 π‘₯ βˆ’ 1 6 6 π‘₯ βˆ’ 9 5 √ 2 1 8 s i n c o s

Q5:

The gradient of the tangent to a curve is βˆ’ π‘₯ + π‘₯ s i n c o s . For π‘₯ ∈ [ 0 , 2 πœ‹ ] , the curve has a local minimum value of 3 5 √ 2 3 9 . Find the equation of the curve.

  • A 𝑦 = π‘₯ + π‘₯ + 7 4 √ 2 3 9 s i n c o s
  • B 𝑦 = π‘₯ + π‘₯ βˆ’ 4 √ 2 3 9 s i n c o s
  • C 𝑦 = βˆ’ π‘₯ βˆ’ π‘₯ + 7 4 √ 2 3 9 s i n c o s
  • D 𝑦 = βˆ’ π‘₯ βˆ’ π‘₯ βˆ’ 4 √ 2 3 9 s i n c o s

Q6:

Find the local minimum value of a curve given that its gradient is d d 𝑦 π‘₯ = π‘₯ + 3 π‘₯ βˆ’ 1 8 2 and the local maximum value is 21.

Q7:

Find the equation of a curve which passes through the point ( 0 , 0 ) and, for each point ( π‘Ž , 𝑏 ) on the curve, the slope of the tangent at that point is βˆ’ 3 π‘₯ √ π‘₯ 5 8 9 .

  • A 𝑦 = βˆ’ 2 7 π‘₯ 6 2 6 2 9
  • B 𝑦 = βˆ’ 8 π‘₯ 1 9 5 7 8
  • C 𝑦 = 6 2 π‘₯ 9 6 2 9
  • D 𝑦 = βˆ’ 6 2 π‘₯ 3 6 2 9

Q8:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is βˆ’ 3 𝑒 6 π‘₯ . What is 𝑓 ( βˆ’ 3 ) , given that 𝑓 ( βˆ’ 5 ) = 9 ?

  • A 9 βˆ’ 1 2 𝑒 + 1 2 𝑒 1 8 3 0
  • B 9 βˆ’ 1 8 𝑒 + 1 2 𝑒 1 8 3 0
  • C 9 βˆ’ 1 8 𝑒 + 1 2 𝑒 3 3 0
  • D 9 βˆ’ 1 2 𝑒 + 1 2 𝑒 3 3 0

Q9:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is βˆ’ 4 𝑒 βˆ’ 5 π‘₯ 7 . What is 𝑓 ( βˆ’ 3 ) , given that 𝑓 ( 9 ) = 9 ?

  • A 2 8 𝑒 5 + 9 βˆ’ 2 8 5 𝑒 1 5 7 4 5 7
  • B 2 0 𝑒 7 + 9 βˆ’ 2 8 5 𝑒 1 5 7 4 5 7
  • C 9 + 2 0 7 𝑒 βˆ’ 2 8 5 𝑒 3 4 5 7
  • D 9 + 2 8 5 𝑒 βˆ’ 2 8 5 𝑒 3 4 5 7

Q10:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is d d s i n c o s 𝑦 π‘₯ = βˆ’ 4 πœ‹ πœ‹ π‘₯ + 5 πœ‹ πœ‹ π‘₯ . Find the equation of the curve if it contains the point ( 1 , 2 ) .

  • A 𝑦 = 5 πœ‹ π‘₯ + 4 πœ‹ π‘₯ + 6 s i n c o s
  • B 𝑦 = 5 πœ‹ πœ‹ π‘₯ + 4 πœ‹ πœ‹ π‘₯ + 6 s i n c o s
  • C 𝑦 = 5 πœ‹ π‘₯ βˆ’ 4 πœ‹ π‘₯ + 6 s i n c o s
  • D 𝑦 = 5 πœ‹ π‘₯ + 4 πœ‹ π‘₯ βˆ’ 2 s i n c o s

Q11:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is d d s i n c o s 𝑦 π‘₯ = 3 πœ‹ πœ‹ π‘₯ βˆ’ 7 πœ‹ πœ‹ π‘₯ . Find the equation of the curve if it contains the point ( 1 , 8 ) .

  • A 𝑦 = βˆ’ 7 πœ‹ π‘₯ βˆ’ 3 πœ‹ π‘₯ + 5 s i n c o s
  • B 𝑦 = βˆ’ 7 πœ‹ πœ‹ π‘₯ βˆ’ 3 πœ‹ πœ‹ π‘₯ + 5 s i n c o s
  • C 𝑦 = βˆ’ 7 πœ‹ π‘₯ + 3 πœ‹ π‘₯ + 5 s i n c o s
  • D 𝑦 = βˆ’ 7 πœ‹ π‘₯ βˆ’ 3 πœ‹ π‘₯ + 1 1 s i n c o s

Q12:

Find the equation of the curve given the gradient of the normal to the curve is √ 2 π‘₯ βˆ’ 2 and the curve passes through the point ( 1 , 6 ) .

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

Q13:

Find the equation of the curve given the gradient of the normal to the curve is √ 8 π‘₯ + 4 and the curve passes through the point ( 4 , 2 ) .

  • A 𝑦 = βˆ’ 1 4 √ 8 π‘₯ + 4 + 7 2
  • B 𝑦 = 1 1 2 √ 8 π‘₯ + 4 + 3 2
  • C 𝑦 = βˆ’ 1 8 √ 8 π‘₯ + 4 + 1 1 4
  • D 𝑦 = 2 √ 8 π‘₯ + 4 βˆ’ 1 0
  • E 𝑦 = βˆ’ 1 1 6 √ 8 π‘₯ + 4 + 1 9 8

Q14:

The second derivative of a curve is βˆ’ 2 7 3 π‘₯ + 8 s i n . The curve passes through the point ο€Ύ πœ‹ 6 , βˆ’ 4 πœ‹ 3 + πœ‹ 9 + 6  2 and the gradient of the tangent at this point is βˆ’ 8 + 4 πœ‹ 3 . Find the equation of the curve.

  • A 𝑦 = 4 π‘₯ βˆ’ 8 π‘₯ + 3 3 π‘₯ + 3 2 s i n
  • B 𝑦 = 4 π‘₯ βˆ’ 8 π‘₯ + 3 3 π‘₯ βˆ’ 3 2 s i n
  • C 𝑦 = 4 π‘₯ βˆ’ 8 π‘₯ + 9 3 π‘₯ βˆ’ 3 2 s i n
  • D 𝑦 = 4 π‘₯ βˆ’ 8 π‘₯ + 9 3 π‘₯ + 3 2 s i n

Q15:

If the rate of change of the sales in a factory is inversely proportional to time in weeks, and the sales of the factory after 2 weeks and 4 weeks are 118 units and 343 units, respectively, determine the sales of the factory after 8 weeks.

Q16:

Find the equation of the curve given the gradient of the tangent is 5 ο€» π‘₯ 2  s i n 2 and the curve passes through the origin.

  • A 𝑦 = 5 2 π‘₯ βˆ’ 5 2 π‘₯ s i n
  • B 𝑦 = 5 ο€» π‘₯ 2  c o s 2
  • C 𝑦 = βˆ’ 5 3 ο€» π‘₯ 2  s i n 3
  • D 𝑦 = 5 π‘₯ βˆ’ 5 π‘₯ s i n

Q17:

The gradient of the tangent to a curve passing through the point is equal to . Find the equation of the tangent at the point when is equal to 1.

  • A
  • B
  • C
  • D

Q18:

A curve passes through ( 1 , 8 ) and the normal at its point ( π‘₯ , 𝑦 ) has slope 8 βˆ’ 9 π‘₯ . What is the equation of the curve?

  • A 𝑦 = 1 9 | 8 βˆ’ 9 π‘₯ | + 8 l n
  • B 𝑦 = βˆ’ 1 9 | 8 βˆ’ 9 π‘₯ | + 8 l n
  • C 𝑦 = βˆ’ 8 π‘₯ + 9 2 π‘₯ + 2 3 2 2
  • D 𝑦 = 8 π‘₯ βˆ’ 9 2 π‘₯ + 9 2 2

Q19:

Given that the slope at ( π‘₯ , 𝑦 ) is 3 𝑒 3 π‘₯ and 𝑓 ( 0 ) = βˆ’ 3 , determine 𝑓 ( βˆ’ 3 ) .

  • A βˆ’ 4 + 1 𝑒 9
  • B βˆ’ 4 + 3 𝑒 9
  • C βˆ’ 4 + 1 𝑒 3
  • D βˆ’ 4 + 9 𝑒 9

Q20:

Given that the slope at ( π‘₯ , 𝑦 ) is βˆ’ 4 𝑒 2 π‘₯ and 𝑓 ( 0 ) = 1 , determine 𝑓 ( 4 ) .

  • A βˆ’ 2 𝑒 + 3 8
  • B βˆ’ 4 𝑒 + 3 8
  • C βˆ’ 2 𝑒 + 3 4
  • D βˆ’ 8 𝑒 + 3 8

Q21:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is 6 𝑒 + 2 π‘₯ . What is 𝑓 ( π‘₯ ) , given that 𝑓 ( 5 ) = 1 l n ?

  • A 6 𝑒 + 2 π‘₯ βˆ’ 2 9 βˆ’ 2 5 π‘₯ l n
  • B 6 𝑒 + 2 π‘₯ + 2 5 + 3 1 π‘₯ l n
  • C 6 𝑒 + 2 π‘₯ + 1 + 2 5 π‘₯ l n
  • D 6 𝑒 + 2 π‘₯ βˆ’ 2 5 + 1 π‘₯ l n

Q22:

A curve passes through the points ο€» πœ‹ 4 , 8  and ο€Ό 3 πœ‹ 4 , βˆ’ 6  . Find the equation of the curve given the gradient of the tangent to the curve equals βˆ’ 7 ( π‘₯ ) c s c 2 .

  • A 𝑦 = 7 π‘₯ + 1 c o t
  • B 𝑦 = 7 π‘₯ + 1 c s c
  • C 𝑦 = βˆ’ 7 π‘₯ + 1 t a n
  • D 𝑦 = βˆ’ 7 π‘₯ + 1 c o t
  • E 𝑦 = βˆ’ 7 π‘₯ + 1 c s c

Q23:

The gradient of the tangent to a curve is d d 𝑦 π‘₯ = π‘₯ βˆ’ 1 4 π‘₯ + 4 5 2 where the value of the local maximum is 9. Find the equation of the curve and the value of the local minimum if it exists.

  • A 𝑦 = π‘₯ 3 βˆ’ 7 π‘₯ + 4 5 π‘₯ βˆ’ 2 4 8 3 3 2 , βˆ’ 5 3
  • B 𝑦 = π‘₯ 3 βˆ’ 7 π‘₯ + 4 5 π‘₯ 3 2 , 9
  • C 𝑦 = π‘₯ βˆ’ 5 π‘₯ + 9 2 , βˆ’ 5 4 5 3
  • D 𝑦 = π‘₯ βˆ’ 9 π‘₯ + 4 5 2 , 5

Q24:

The slope at the point ( π‘₯ , 𝑓 ( π‘₯ ) ) on the graph of a function is 4 βˆ’ 5 𝑒 + 4 π‘₯ . What is 𝑓 ( 4 𝑒 ) if we know that 𝑓 ( 𝑒 ) = βˆ’ 9 ?

  • A l n 1 1 𝑒 βˆ’ 1 0
  • B 4 1 6 𝑒 βˆ’ 6 6 5 l n
  • C βˆ’ 1 0 + 1 1 1 𝑒 l n
  • D βˆ’ 6 4 𝑒 5 βˆ’ 1 0 + 4 1 6 𝑒 l n

Q25:

The slope at the point ( π‘₯ , 𝑦 ) on the graph of a function is 5 π‘₯ βˆ’ 2 π‘₯ . Find the equation of the curve if it contains the point ( 𝑒 , 5 𝑒 + 3 ) .

  • A 5 π‘₯ βˆ’ 2 | π‘₯ | + 5 l n
  • B 5 π‘₯ + 5 βˆ’ 2 | π‘₯ | l n
  • C 1 0 𝑒 + 5 π‘₯ βˆ’ 2 | π‘₯ | + 1 l n
  • D 5 π‘₯ βˆ’ 2 | π‘₯ | + 1 l n
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