Lesson Worksheet: Equations of Tangent Lines and Normal Lines Mathematics • Higher Education

In this worksheet, we will practice finding the slope and equation of the tangent and normal to a curve at a given point using derivatives.


Find the equation of the tangent to the curve 𝑦=βˆ’2π‘₯+8π‘₯βˆ’19 at π‘₯=2.

  • Aβˆ’8𝑦+π‘₯βˆ’2=0
  • Bπ‘¦βˆ’8π‘₯+19=0
  • C𝑦+8π‘₯+19=0
  • Dπ‘¦βˆ’4π‘₯βˆ’5=0


Find the equation of the normal to the curve 𝑦=βˆ’2π‘₯βˆ’7π‘₯+2 at π‘₯=βˆ’2.

  • A𝑦+2π‘₯+6=0
  • B4𝑦+π‘₯+42=0
  • C𝑦+4π‘₯+2=0
  • D𝑦+6π‘₯βˆ’2=0


What is the π‘₯-coordinate of the point where the tangent line to 𝑦=π‘₯+12π‘₯+11 is parallel to the π‘₯-axis?

  • Aβˆ’12
  • B0
  • C6
  • Dβˆ’6


Find the equation of the tangent to the curve 𝑦=π‘₯+9π‘₯+26π‘₯ that makes an angle of 135∘ with the positive π‘₯-axis.

  • Aπ‘¦βˆ’π‘₯3+23=0
  • B𝑦+27π‘₯+105=0
  • C𝑦+π‘₯+27=0
  • Dπ‘¦βˆ’8π‘₯=0


Find all points with π‘₯-coordinates in [0,πœ‹) where the curve 𝑦=2π‘₯sin has a tangent that is parallel to the line 𝑦=βˆ’π‘₯βˆ’18.

  • Aο€Ώπœ‹3,√32,ο€Ώ2πœ‹3,√32
  • Bο€Όπœ‹3,βˆ’12,ο€Ό2πœ‹3,βˆ’12
  • Cο€Ώπœ‹3,√32,ο€Ώ2πœ‹3,βˆ’βˆš32
  • Dο€Όβˆ’πœ‹3,12,ο€Όβˆ’2πœ‹3,12


Find the equations to the tangent lines of the curve 𝑦=(π‘₯+8)(π‘₯+10) at the points where this curve intersects the π‘₯-axis.

  • A𝑦+2π‘₯+16=0, π‘¦βˆ’2π‘₯βˆ’20=0
  • Bπ‘¦βˆ’2π‘₯βˆ’16=0, 𝑦+2π‘₯+20=0
  • C𝑦+2π‘₯βˆ’16=0, π‘¦βˆ’2π‘₯+20=0
  • Dπ‘¦βˆ’2π‘₯+16=0, 𝑦+2π‘₯βˆ’20=0


Find the equation of the tangent to the curve 𝑓(π‘₯)=π‘₯ at its point of intersection with the curve 𝑔(π‘₯)=125π‘₯.

  • Aπ‘¦βˆ’10π‘₯+25=0
  • B𝑦+10π‘₯βˆ’25=0
  • C10𝑦+π‘₯βˆ’255=0
  • D10π‘¦βˆ’π‘₯βˆ’245=0


Find the equation of the normal to the curve 𝑦=5π‘₯+93π‘₯βˆ’5 at (1,βˆ’7).

  • Aβˆ’13π‘¦βˆ’π‘₯βˆ’90=0
  • B13π‘¦βˆ’π‘₯+92=0
  • C𝑦+13π‘₯βˆ’6=0
  • Dπ‘¦βˆ’13π‘₯+20=0


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

  • Aο€Ό13,499, ο€Όβˆ’13,779
  • Bο€Ώβˆš33,499, ο€Ώβˆ’βˆš33,4√33+7
  • Cο€Ό13,23, ο€Όβˆ’13,103
  • Dο€Ό16,499, ο€Όβˆ’16,56372


The curves 𝑦=2π‘₯βˆ’3π‘₯βˆ’2 and 𝑦=βˆ’3π‘₯+5π‘₯βˆ’5 intersect orthogonally at a point. What is this point?

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

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