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Worksheet: Applications of the Indefinite Integration

Q1:

A curve passes through and the tangent at its point has gradient . What is the equation of the curve?

  • A
  • B
  • C
  • D

Q2:

A curve passes through and the tangent at its point has gradient . What is the equation of the curve?

  • A
  • B
  • C
  • D

Q3:

Find the equation of the curve that passes through the point ( 2 , 1 ) given that the gradient of the tangent to the curve is 1 1 𝑥 2 .

  • A 𝑦 = 1 1 3 𝑥 + 3 C
  • B 𝑦 = 1 1 3 𝑥 + 4 7 3 2
  • C 𝑦 = 1 1 𝑥 + 9 3
  • D 𝑦 = 1 1 3 𝑥 8 5 3 3

Q4:

The gradient of the tangent to a curve is . For , the curve has a local minimum value of . Find the equation of the curve.

  • A
  • B
  • C
  • D

Q5:

The gradient of the tangent to a curve is . For , the curve has a local minimum value of . Find the equation of the curve.

  • A
  • B
  • C
  • D

Q6:

Find the local minimum value of a curve given that its gradient is d d 𝑦 𝑥 = 𝑥 + 3 𝑥 1 8 2 and the local maximum value is 21.

Q7:

Find the equation of a curve which passes through the point and, for each point on the curve, the gradient of the tangent at that point is .

  • A
  • B
  • C
  • D

Q8:

The gradient at the point on the graph of a function is . What is , given that ?

  • A
  • B
  • C
  • D

Q9:

The gradient at the point on the graph of a function is . What is , given that ?

  • A
  • B
  • C
  • D

Q10:

The gradient at the point on the graph of a function is . Find the equation of the curve if it contains the point .

  • A
  • B
  • C
  • D

Q11:

The gradient at the point on the graph of a function is . Find the equation of the curve if it contains the point .

  • A
  • B
  • C
  • D

Q12:

Find the equation of the curve given the gradient of the normal to the curve is 2 𝑥 2 and the curve passes through the point ( 1 , 6 ) .

  • A 𝑦 = 1 4 2 𝑥 2 + 6
  • B 𝑦 = 1 2 2 𝑥 2 + 6
  • C 𝑦 = 2 2 𝑥 2 + 6
  • D 𝑦 = 2 𝑥 2 + 6
  • E 𝑦 = 1 3 2 𝑥 2 + 6

Q13:

Find the equation of the curve given the gradient of the normal to the curve is 8 𝑥 + 4 and the curve passes through the point ( 4 , 2 ) .

  • A 𝑦 = 1 1 6 8 𝑥 + 4 + 1 9 8
  • B 𝑦 = 1 8 8 𝑥 + 4 + 1 1 4
  • C 𝑦 = 2 8 𝑥 + 4 1 0
  • D 𝑦 = 1 4 8 𝑥 + 4 + 7 2
  • E 𝑦 = 1 1 2 8 𝑥 + 4 + 3 2

Q14:

The second derivative of a curve is 2 7 3 𝑥 + 8 s i n . The curve passes through the point 𝜋 6 , 4 𝜋 3 + 𝜋 9 + 6 2 and the gradient of the tangent at this point is 8 + 4 𝜋 3 . Find the equation of the curve.

  • A 𝑦 = 4 𝑥 8 𝑥 + 9 3 𝑥 + 3 2 s i n
  • B 𝑦 = 4 𝑥 8 𝑥 + 3 3 𝑥 3 2 s i n
  • C 𝑦 = 4 𝑥 8 𝑥 + 9 3 𝑥 3 2 s i n
  • D 𝑦 = 4 𝑥 8 𝑥 + 3 3 𝑥 + 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 𝑥 2 s i n 2 and the curve passes through the origin.

  • A 𝑦 = 5 𝑥 5 𝑥 s i n
  • B 𝑦 = 5 𝑥 2 c o s 2
  • C 𝑦 = 5 3 𝑥 2 s i n 3
  • D 𝑦 = 5 2 𝑥 5 2 𝑥 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 and the normal at its point has gradient . What is the equation of the curve?

  • A
  • B
  • C
  • D

Q19:

Given that the gradient at is and , determine .

  • A
  • B
  • C
  • D

Q20:

Given that the gradient at is and , determine .

  • A
  • B
  • C
  • D

Q21:

The gradient at the point on the graph of a function is . What is , given that ?

  • A
  • B
  • C
  • D

Q22:

A curve passes through the points 𝜋 4 , 8 and 3 𝜋 4 , 6 . Find the equation of the curve given the gradient of the tangent to the curve equals 7 ( 𝑥 ) c s c 2 .

  • A 𝑦 = 7 𝑥 + 1 c s c
  • B 𝑦 = 7 𝑥 + 1 t a n
  • C 𝑦 = 7 𝑥 + 1 c o t
  • D 𝑦 = 7 𝑥 + 1 c o t
  • E 𝑦 = 7 𝑥 + 1 c s c

Q23:

The gradient of the tangent to a curve is d d 𝑦 𝑥 = 𝑥 1 4 𝑥 + 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 = 𝑥 9 𝑥 + 4 5 2 , 5
  • B 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 3 2 , 9
  • C 𝑦 = 𝑥 5 𝑥 + 9 2 , 5 4 5 3
  • D 𝑦 = 𝑥 3 7 𝑥 + 4 5 𝑥 2 4 8 3 3 2 , 5 3

Q24:

The gradient at the point on the graph of a function is . What is if we know that ?

  • A
  • B
  • C
  • D

Q25:

The gradient at the point on the graph of a function is . Find the equation of the curve if it contains the point .

  • A
  • B
  • C