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.

Q1:

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

Q2:

If the line 𝑦=3π‘₯+9 is tangent to the graph of the function 𝑓 at (2,15), what is 𝑓′(2)?

Q3:

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

Q4:

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

Q5:

Determine the equation of the line tangent to the curve 𝑦=4π‘₯βˆ’2π‘₯+4 at point (βˆ’1,βˆ’2).

  • A𝑦=8π‘₯+6
  • B𝑦=16π‘₯+16
  • C𝑦=14π‘₯+12
  • D𝑦=16π‘₯+14
  • E𝑦=16π‘₯βˆ’2

Q6:

The line 𝑦=5π‘₯+4 is tangent to the graph of function 𝑓 at the point (βˆ’1,βˆ’1). What is 𝑓′(βˆ’1)?

Q7:

Find the equation of the tangent to the curve 𝑦=π‘₯βˆ’2π‘₯ at the point (π‘₯,3) on the curve.

  • Aπ‘¦βˆ’4π‘₯+9=0, 𝑦+4π‘₯+1=0
  • B4π‘¦βˆ’π‘₯βˆ’9=0, βˆ’4π‘¦βˆ’π‘₯+11=0
  • Cβˆ’4π‘¦βˆ’π‘₯+15=0, 4π‘¦βˆ’π‘₯βˆ’13=0
  • D𝑦+4π‘₯βˆ’15=0, π‘¦βˆ’4π‘₯βˆ’7=0

Q8:

List the equations of the normals to 𝑦=π‘₯+2π‘₯ at the points where this curve meets the line π‘¦βˆ’4π‘₯=0.

  • A2𝑦+π‘₯=0, 6𝑦+π‘₯βˆ’50=0
  • B𝑦+2π‘₯=0, 𝑦+6π‘₯βˆ’4=0
  • Cπ‘¦βˆ’2π‘₯=0, π‘¦βˆ’6π‘₯+4=0
  • D2π‘¦βˆ’π‘₯=0, 6π‘¦βˆ’π‘₯βˆ’46=0

Q9:

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

Q10:

Find the equation of the normal to the curve 𝑦=π‘₯tan at π‘₯=πœ‹4.

  • Aβˆ’14π‘₯+7π‘¦βˆ’4=0
  • B28𝑦+14π‘₯βˆ’39=0
  • C28π‘₯+14π‘¦βˆ’39=0
  • D14π‘₯βˆ’7π‘¦βˆ’4=0

Q11:

Find the equations of the two tangents to the curve 𝑦=π‘₯+6π‘₯βˆ’6 that are perpendicular to the line π‘₯+9𝑦=9.

  • Aβˆ’9π‘¦βˆ’π‘₯+8=0, βˆ’9π‘¦βˆ’π‘₯βˆ’116=0
  • Bπ‘¦βˆ’9π‘₯+8=0, π‘¦βˆ’9π‘₯+4=0
  • C𝑦+9π‘₯βˆ’10=0, 𝑦+9π‘₯+22=0
  • D9π‘¦βˆ’π‘₯βˆ’10=0, 9π‘¦βˆ’π‘₯+118=0

Q12:

Find the equations of the tangents to the curve 𝑦=βˆ’π‘₯+4π‘₯βˆ’18 that are parallel to the straight line βˆ’π‘₯+𝑦=βˆ’3.

  • Aπ‘¦βˆ’π‘₯βˆ’16=0, π‘¦βˆ’π‘₯βˆ’20=0
  • Bπ‘¦βˆ’π‘₯βˆ’16=0, π‘¦βˆ’π‘₯+20=0
  • Cπ‘¦βˆ’π‘₯+16=0, π‘¦βˆ’π‘₯βˆ’20=0
  • Dπ‘¦βˆ’π‘₯+16=0, π‘¦βˆ’π‘₯+20=0

Q13:

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

Q14:

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

Q15:

Determine the equation of the tangent to the curve 𝑦=βˆ’π‘₯βˆ’3π‘₯ at point (βˆ’1,4).

  • A𝑦=2π‘₯+6
  • B𝑦=βˆ’2π‘₯βˆ’6
  • C𝑦=3π‘₯+7
  • D𝑦=3π‘₯+1
  • E𝑦=2π‘₯βˆ’2

Q16:

Find the equation of the tangent to the curve 𝑦=π‘₯|π‘₯|+2π‘₯ at π‘₯=βˆ’3.

  • A8𝑦+π‘₯+123=0
  • Bπ‘¦βˆ’8π‘₯βˆ’9=0
  • C8π‘¦βˆ’π‘₯+117=0
  • D𝑦+4π‘₯+9=0
  • E𝑦+8π‘₯+9=0

Q17:

Find the equation of the tangent to the curve 𝑦=2π‘₯βˆ’45π‘₯sincos at π‘₯=3πœ‹2.

  • Aβˆ’22𝑦+π‘₯βˆ’3πœ‹2=0
  • B𝑦+22π‘₯+33πœ‹=0
  • C𝑦+22π‘₯βˆ’33πœ‹=0
  • Dπ‘¦βˆ’22π‘₯+33πœ‹=0

Q18:

Find the equation of the tangent to the graph of 𝑓(π‘₯)=βˆ’7π‘₯+44π‘₯tan at the point ο€»πœ‹2,π‘“ο€»πœ‹2.

  • A𝑦+9π‘₯+8πœ‹=0
  • B𝑦+9π‘₯βˆ’8πœ‹=0
  • Cπ‘¦βˆ’9π‘₯βˆ’8πœ‹=0
  • Dπ‘¦βˆ’9π‘₯+8πœ‹=0

Q19:

Find the equation of the normal to 𝑦=54π‘₯+7π‘₯sincos at π‘₯=πœ‹2.

  • A13𝑦+π‘₯+πœ‹2=0
  • B13π‘¦βˆ’π‘₯+πœ‹2=0
  • Cπ‘¦βˆ’13π‘₯+13πœ‹2=0
  • D13𝑦+π‘₯βˆ’πœ‹2=0

Q20:

Find the equation of the tangent to the curve 𝑦=43π‘₯sin at π‘₯=πœ‹12, giving your answer in terms of πœ‹.

  • A12π‘¦βˆ’π‘₯+πœ‹12βˆ’24=0
  • Bπ‘¦βˆ’12π‘₯+πœ‹βˆ’2=0
  • C𝑦+12π‘₯βˆ’πœ‹βˆ’2=0
  • Dβˆ’12π‘¦βˆ’π‘₯+πœ‹12+24=0

Q21:

Find the equation of the normal to the curve 𝑦=9π‘₯+29π‘₯tan at π‘₯=0.

  • Aβˆ’9π‘₯+𝑦=0
  • Bπ‘₯+27𝑦=0
  • C27π‘₯+𝑦=0
  • Dβˆ’27π‘₯+𝑦=0

Q22:

Find the equation of the normal to the curve 𝑦=√3π‘₯+33π‘₯sincos at π‘₯=πœ‹6.

  • A92π‘¦βˆ’π‘₯+ο€Όπœ‹6βˆ’92=0
  • B𝑦+92π‘₯βˆ’ο€Ό3πœ‹4+1=0
  • Cπ‘¦βˆ’92π‘₯+ο€Ό3πœ‹4βˆ’1=0
  • Dβˆ’92π‘¦βˆ’π‘₯+ο€Όπœ‹6+92=0

Q23:

Find the equation of the normal to the curve 𝑦=8π‘₯sin at π‘₯=πœ‹4.

  • Aπ‘¦βˆ’8π‘₯+2πœ‹βˆ’4=0
  • B8π‘¦βˆ’π‘₯+πœ‹4βˆ’32=0
  • Cβˆ’8π‘¦βˆ’π‘₯+πœ‹4+32=0
  • D𝑦+8π‘₯βˆ’2πœ‹+4=0

Q24:

Find the equation of the tangent to the curve 𝑦=6+9π‘₯3βˆ’9π‘₯sinsin at the point (πœ‹,2).

  • A𝑦+27π‘₯βˆ’(27πœ‹+2)=0
  • B𝑦+9π‘₯βˆ’(9πœ‹+2)=0
  • Cπ‘¦βˆ’9π‘₯+(9πœ‹βˆ’2)=0
  • D𝑦+81π‘₯βˆ’(81πœ‹+2)=0
  • E𝑦+3π‘₯βˆ’(3πœ‹+2)=0

Q25:

Find the equation of the tangent to the curve 𝑦=√93π‘₯+43π‘₯sincos at π‘₯=πœ‹6.

  • Aβˆ’2π‘¦βˆ’π‘₯+ο€»πœ‹6+6=0
  • Bπ‘¦βˆ’2π‘₯+ο€»πœ‹3βˆ’3=0
  • C𝑦+2π‘₯βˆ’ο€»πœ‹3+3=0
  • D2π‘¦βˆ’π‘₯+ο€»πœ‹6βˆ’6=0

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