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Worksheet: Slopes of Tangents to Curves

Q1:

Let Suppose that the tangent to the graph of 𝑓 at π‘₯ = 1 makes an angle with the positive π‘₯ -axis of tangent βˆ’ 7 . Find π‘Ž and 𝑏 .

  • A π‘Ž = βˆ’ 1 3 7 , 𝑏 = 1 7
  • B π‘Ž = 9 , 𝑏 = βˆ’ 7
  • C π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 7
  • D π‘Ž = βˆ’ 9 , 𝑏 = βˆ’ 7

Q2:

Find the points on the curve 𝑦 = 3 π‘₯ βˆ’ 5 π‘₯ + 7 3 at which the tangents are parallel to the line 4 π‘₯ + 𝑦 βˆ’ 2 = 0 .

  • A ο€Ό 1 6 , 4 9 9  , ο€Ό βˆ’ 1 6 , 5 6 3 7 2 
  • B ο€Ό 1 3 , 2 3  , ο€Ό βˆ’ 1 3 , 1 0 3 
  • C ο€Ώ √ 3 3 , 4 9 9  , ο€Ώ βˆ’ √ 3 3 , 4 √ 3 3 + 7 
  • D ο€Ό 1 3 , 4 9 9  , ο€Ό βˆ’ 1 3 , 7 7 9 

Q3:

Let be the tangent to the curve at the point . Determine the gradient of and the angle it makes with the positive -axis, giving your answer to the nearest minute.

  • AThe gradient of is 8, and the angle it makes with the positive -axis is .
  • BThe gradient of is 4, and the angle it makes with the positive -axis is .
  • CThe gradient of is 2, and the angle it makes with the positive -axis is .
  • DThe gradient of is 8, and the angle it makes with the positive -axis is .
  • EThe gradient of is 16, and the angle it makes with the positive -axis is .

Q4:

Find the points on the curve where the tangent has gradient 4.

  • A ,
  • B ,
  • C ,
  • D ,

Q5:

Determine the gradient of the tangent to the curve of the function when .

  • A5
  • B
  • C
  • D

Q6:

Find the points on the curve where the gradient of the tangent line is .

  • A
  • B
  • C
  • D

Q7:

Let be the tangent to the curve at the point . Determine the gradient of and the angle it makes with the positive -axis, giving your answer to the nearest minute.

  • AThe gradient of is 3, and the angle it makes with the positive -axis is .
  • BThe gradient of is 21, and the angle it makes with the positive -axis is .
  • CThe gradient of is 7, and the angle it makes with the positive -axis is .
  • DThe gradient of is 21, and the angle it makes with the positive -axis is .

Q8:

Find the points on 𝑦 = π‘₯ βˆ’ 6 π‘₯ + 1 0 π‘₯ + 1 0 3 2 where the tangent is parallel to the line that passes through ( 0 , 5 ) and ( βˆ’ 6 , βˆ’ 1 ) .

  • A ( 1 , 1 5 ) and ( βˆ’ 3 , 1 3 )
  • B ( βˆ’ 1 , 1 5 ) and ( βˆ’ 3 , 1 3 )
  • C ( βˆ’ 1 , 1 5 ) and ( 3 , 1 3 )
  • D ( 1 , 1 5 ) and ( 3 , 1 3 )

Q9:

Find the points on the curve 𝑦 = ( π‘₯ βˆ’ 3 ) βˆ’ 9 2 where the tangent is parallel to the line βˆ’ 6 π‘₯ + 𝑦 βˆ’ 2 4 = 0 .

  • A ( 6 , βˆ’ 6 )
  • B ( 9 , 0 )
  • C ( 6 , 3 6 )
  • D ( 6 , 0 )
  • E ο€Ό 3 5 1 2 , 0 

Q10:

Find the gradient of the tangent to at the point where .

  • A
  • B81
  • C26
  • D86

Q11:

Find the gradient of the tangent to at the point where .

  • A95
  • B
  • C
  • D

Q12:

Find the gradient of the tangent to at the point where .

  • A10
  • B
  • C
  • D

Q13:

At which points are tangents to the curve 𝑦 = βˆ’ π‘₯ βˆ’ 6 π‘₯ βˆ’ 9 π‘₯ + 9 3 2 parallel to the π‘₯ -axis?

  • A ( 0 , 9 ) and ( 1 , βˆ’ 7 )
  • B ( βˆ’ 3 , 9 )
  • C ( 0 , 9 )
  • D ( βˆ’ 3 , 9 ) and ( βˆ’ 1 , 1 3 )

Q14:

Find the point that lies on the curve 𝑦 = βˆ’ π‘₯ βˆ’ 8 π‘₯ βˆ’ 9 2 , at which the tangent to the curve is perpendicular to the straight line 4 𝑦 + π‘₯ + 7 = 0 .

  • A ο€Ό βˆ’ 7 2 , 2 7 4 
  • B ( βˆ’ 4 , 7 )
  • C ο€Ό βˆ’ 6 , βˆ’ 1 4 
  • D ( βˆ’ 6 , 3 )

Q15:

Find the points on the curve 𝑦 = π‘₯ βˆ’ 5 π‘₯ βˆ’ 8 3 where the tangent is perpendicular to the line π‘₯ + 7 𝑦 βˆ’ 9 = 0 .

  • A ο€» √ 2 , βˆ’ 8 βˆ’ 3 √ 2  , ο€» βˆ’ √ 2 , βˆ’ 8 + 3 √ 2 
  • B ο€Ώ √ 6 3 , βˆ’ 8 βˆ’ 1 3 √ 6 9  , ο€Ώ βˆ’ √ 6 3 , βˆ’ 8 + 1 3 √ 6 9 
  • C ο€» 2 √ 3 , βˆ’ 8 + 1 4 √ 3  , ο€» βˆ’ 2 √ 3 , βˆ’ 1 4 √ 3 βˆ’ 8 
  • D ( 2 , βˆ’ 1 0 ) , ( βˆ’ 2 , βˆ’ 6 )

Q16:

Find the gradient of the tangent to the curve at the point .

  • A
  • B
  • C
  • D18

Q17:

Find the gradient of the tangent to the curve at the point .

  • A
  • B
  • C17
  • D5

Q18:

Find the gradient of the tangent to the curve at the point .

  • A
  • B
  • C11
  • D20

Q19:

Find the gradient of the tangent to the curve at .

  • A12
  • B8
  • C7
  • D16

Q20:

Find the gradient of the tangent to the curve at .

  • A23
  • B7
  • C3
  • D39

Q21:

Find the gradient of the tangent to the curve at .

  • A63
  • B31
  • C11
  • D95

Q22:

Find the gradient of the tangent to the curve at .

  • A12
  • B3
  • C
  • D21

Q23:

Differentiate , and find the gradient of the tangent to its graph at .

  • A , gradient of the tangent to the graph
  • B , gradient of the tangent to the graph
  • C , gradient of the tangent to the graph
  • D , gradient of the tangent to the graph

Q24:

Differentiate , and find the gradient of the tangent to its graph at .

  • A , gradient of the tangent to the graph
  • B , gradient of the tangent to the graph
  • C , gradient of the tangent to the graph
  • D , gradient of the tangent to the graph

Q25:

Differentiate , and find the gradient of the tangent to its graph at .

  • A , gradient of the tangent to the graph
  • B , gradient of the tangent to the graph
  • C , gradient of the tangent to the graph
  • D , gradient of the tangent to the graph