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Lesson: Slopes of Tangents to Curves

Worksheet • 25 Questions

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’ 9 , 𝑏 = βˆ’ 7
  • B π‘Ž = 9 , 𝑏 = βˆ’ 7
  • C π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 7
  • D π‘Ž = βˆ’ 1 3 7 , 𝑏 = 1 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 3 , 4 9 9  , ο€Ό βˆ’ 1 3 , 7 7 9 
  • 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 6 , 4 9 9  , ο€Ό βˆ’ 1 6 , 5 6 3 7 2 

Q3:

Let 𝐿 be the tangent to the curve 𝑦 = 4 π‘₯ + 5 2 at the point ( 1 , 9 ) . Determine the slope of 𝐿 and the angle it makes with the positive π‘₯ -axis, giving your answer to the nearest minute.

  • AThe slope of 𝐿 is 8, and the angle it makes with the positive π‘₯ -axis is 8 2 5 2 ∘ β€² .
  • BThe slope of 𝐿 is 16, and the angle it makes with the positive π‘₯ -axis is 8 6 2 5 ∘ β€² .
  • CThe slope of 𝐿 is 4, and the angle it makes with the positive π‘₯ -axis is 7 5 5 8 ∘ β€² .
  • DThe slope of 𝐿 is 2, and the angle it makes with the positive π‘₯ -axis is 6 3 2 6 ∘ β€² .
  • EThe slope of 𝐿 is 8, and the angle it makes with the positive π‘₯ -axis is 8 2 5 2 ∘ β€² .

Q4:

Find the points on the curve 𝑦 = 3 π‘₯ + 5 π‘₯ βˆ’ 8 3 where the tangent has slope 4.

  • A ( 1 , 0 ) , ( βˆ’ 1 , βˆ’ 1 6 )
  • B ( 1 . 7 3 , 1 6 . 2 5 ) , ( βˆ’ 1 . 7 3 , βˆ’ 3 2 . 2 5 )
  • C ( 1 , 4 ) , ( βˆ’ 1 , 4 )
  • D ( 1 . 2 2 , 3 . 6 4 ) , ( βˆ’ 1 . 2 2 , βˆ’ 1 9 . 6 4 )

Q5:

Determine the slope of the tangent to the curve of the function 𝑦 = ( βˆ’ π‘₯ + 1 ) 5 when π‘₯ = 0 .

Q6:

Find the points on the curve 𝑦 = ( π‘₯ βˆ’ 3 ) ο€Ή π‘₯ βˆ’ 2  2 where the slope of the tangent line is βˆ’ 2 .

  • A ( 2 , βˆ’ 2 ) , ( 0 , 6 )
  • B ( 0 , βˆ’ 2 ) , ( βˆ’ 3 , 6 )
  • C ( 2 , 6 ) , ( 0 , βˆ’ 2 )
  • D ( 2 , βˆ’ 7 ) , ( βˆ’ 3 , 1 )

Q7:

Let 𝐿 be the tangent to the curve 𝑦 = 7 π‘₯ + 2 3 at the point ( 1 , 9 ) . Determine the slope of 𝐿 and the angle it makes with the positive π‘₯ -axis, giving your answer to the nearest minute.

  • AThe slope of 𝐿 is 21, and the angle it makes with the positive π‘₯ -axis is 8 7 1 6 β€² ∘ .
  • BThe slope of 𝐿 is 21, and the angle it makes with the positive π‘₯ -axis is 8 7 1 6 β€² ∘ .
  • CThe slope of 𝐿 is 7, and the angle it makes with the positive π‘₯ -axis is 8 1 5 2 β€² ∘ .
  • DThe slope of 𝐿 is 3, and the angle it makes with the positive π‘₯ -axis is 7 1 3 4 β€² ∘ .

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

Q10:

Find the slope of the tangent to βˆ’ 𝑦 = 8 π‘₯ βˆ’ 6 π‘₯ + 5 2 at the point where π‘₯ = βˆ’ 5 .

Q11:

Find the slope of the tangent to 𝑦 = 2 π‘₯ βˆ’ 8 π‘₯ + 5 2 at the point where π‘₯ = βˆ’ 5 .

Q12:

Find the slope of the tangent to 𝑦 = 6 π‘₯ + 3 π‘₯ βˆ’ 8 2 at the point where π‘₯ = βˆ’ 2 .

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

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

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 ( 2 , βˆ’ 1 0 ) , ( βˆ’ 2 , βˆ’ 6 )
  • 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 ο€» √ 2 , βˆ’ 8 βˆ’ 3 √ 2  , ο€» βˆ’ √ 2 , βˆ’ 8 + 3 √ 2 

Q16:

Find the slope of the tangent to the curve 𝑦 = 9 π‘₯ βˆ’ 9 π‘₯ βˆ’ 8 3 at the point ( 1 , βˆ’ 8 ) .

Q17:

Find the slope of the tangent to the curve 𝑦 = π‘₯ βˆ’ 7 π‘₯ + 1 3 at the point ( 2 , βˆ’ 5 ) .

Q18:

Find the slope of the tangent to the curve 𝑦 = 8 π‘₯ βˆ’ 4 π‘₯ + 7 3 at the point ( 1 , 1 1 ) .

Q19:

Find the slope of the tangent to the curve 𝑦 = 4 π‘₯ + 4 π‘₯ 3 at π‘₯ = 1 .

Q20:

Find the slope of the tangent to the curve 𝑦 = 4 π‘₯ βˆ’ 9 π‘₯ 3 at π‘₯ = 2 .

Q21:

Find the slope of the tangent to the curve 𝑦 = 8 π‘₯ βˆ’ π‘₯ 3 at π‘₯ = 2 .

Q22:

Find the slope of the tangent to the curve 𝑦 = 9 π‘₯ βˆ’ 6 π‘₯ 3 at π‘₯ = βˆ’ 1 .

Q23:

Differentiate 𝑓 ( π‘₯ ) = 3 π‘₯ + 2 8 , and find the slope of the tangent to its graph at ( 1 , 5 ) .

  • A 𝑓 β€² ( π‘₯ ) = 2 4 π‘₯ 7 , slope of the tangent to the graph = 2 4
  • B 𝑓 β€² ( π‘₯ ) = 3 π‘₯ 7 , slope of the tangent to the graph = 3
  • C 𝑓 β€² ( π‘₯ ) = 3 π‘₯ 9 , slope of the tangent to the graph = 3
  • D 𝑓 β€² ( π‘₯ ) = 2 4 π‘₯ 9 , slope of the tangent to the graph = 2 4

Q24:

Differentiate 𝑓 ( π‘₯ ) = βˆ’ 5 π‘₯ + 6 4 , and find the slope of the tangent to its graph at ( βˆ’ 1 , 1 ) .

  • A 𝑓 β€² ( π‘₯ ) = βˆ’ 2 0 π‘₯ 3 , slope of the tangent to the graph = 2 0
  • B 𝑓 β€² ( π‘₯ ) = βˆ’ 5 π‘₯ 3 , slope of the tangent to the graph = 5
  • C 𝑓 β€² ( π‘₯ ) = βˆ’ 5 π‘₯ 5 , slope of the tangent to the graph = 5
  • D 𝑓 β€² ( π‘₯ ) = βˆ’ 2 0 π‘₯ 5 , slope of the tangent to the graph = 2 0

Q25:

Differentiate 𝑓 ( π‘₯ ) = 4 π‘₯ + 5 5 , and find the slope of the tangent to its graph at ( βˆ’ 1 , 1 ) .

  • A 𝑓 β€² ( π‘₯ ) = 2 0 π‘₯ 4 , slope of the tangent to the graph = 2 0
  • B 𝑓 β€² ( π‘₯ ) = 4 π‘₯ 4 , slope of the tangent to the graph = 4
  • C 𝑓 β€² ( π‘₯ ) = 4 π‘₯ 6 , slope of the tangent to the graph = 4
  • D 𝑓 β€² ( π‘₯ ) = 2 0 π‘₯ 6 , slope of the tangent to the graph = 2 0
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