Worksheet: Tangents to the Graph of a Function

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

Q1:

Find the equation of the tangent to the curve 𝑦 = 2 𝑥 + 8 𝑥 1 9 at 𝑥 = 2 .

  • A 𝑦 4 𝑥 5 = 0
  • B 8 𝑦 + 𝑥 2 = 0
  • C 𝑦 + 8 𝑥 + 1 9 = 0
  • D 𝑦 8 𝑥 + 1 9 = 0

Q2:

If the line 𝑦 = 3 𝑥 + 9 is tangent to the graph of the function 𝑓 at ( 2 , 1 5 ) , what is 𝑓 ( 2 ) ?

Q3:

What is the 𝑥 -coordinate of the point where the tangent line to 𝑦 = 𝑥 + 1 2 𝑥 + 1 1 is parallel to the 𝑥 -axis?

  • A 1 2
  • B6
  • C0
  • D 6

Q4:

The point ( 3 , 3 ) lies on the curve 𝑦 = 7 𝑥 + 𝑎 𝑥 + 𝑏 . If the slope of the tangent there is 1 , what are the values of constants 𝑎 and 𝑏 ?

  • A 𝑎 = 4 3 , 𝑏 = 1 7
  • B 𝑎 = 4 1 , 𝑏 = 6 3
  • C 𝑎 = 4 1 , 𝑏 = 1 8 3
  • D 𝑎 = 4 3 , 𝑏 = 6 9

Q5:

Find the point on the curve 𝑦 = 4 0 𝑥 + 4 0 at which the tangent to the curve is parallel to the 𝑥 -axis.

  • A ( 1 , 0 )
  • B ( 0 , 0 )
  • C ( 1 , 0 )
  • D ( 0 , 4 0 )

Q6:

List the equations of all the tangents to 𝑦 = 𝑥 that also lie on the point ( 2 , 3 ) .

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

Q7:

Find the equation of the tangent to the curve 𝑦 = 𝑥 + 9 𝑥 + 2 6 𝑥 that makes an angle of 1 3 5 with the positive 𝑥 -axis.

  • A 𝑦 + 2 7 𝑥 + 1 0 5 = 0
  • B 𝑦 8 𝑥 = 0
  • C 𝑦 𝑥 3 + 2 3 = 0
  • D 𝑦 + 𝑥 + 2 7 = 0

Q8:

Suppose the line 𝑦 + 5 𝑥 1 = 0 touches the curve 𝑓 ( 𝑥 ) = 𝑥 𝑥 + 𝑎 . What is 𝑎 ?

Q9:

If the curve 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 + 2 𝑥 + 7 is tangent to the line 𝑦 = 7 𝑥 3 at ( 1 , 1 0 ) , find the constants 𝑎 and 𝑏 .

  • A 𝑎 = 2 5 , 𝑏 = 4 0
  • B 𝑎 = 4 0 , 𝑏 = 2 5
  • C 𝑎 = 5 , 𝑏 = 1 0
  • D 𝑎 = 2 5 , 𝑏 = 4 0

Q10:

The line 𝑥 𝑦 3 = 0 touches the curve 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 at ( 1 , 2 ) . Find 𝑎 and 𝑏 .

  • A 𝑎 = 1 3 , 𝑏 = 7
  • B 𝑎 = 7 , 𝑏 = 5
  • C 𝑎 = 1 3 , 𝑏 = 1 5
  • D 𝑎 = 5 , 𝑏 = 7

Q11:

Do the curves 𝑦 = 2 𝑥 + 4 𝑥 + 2 4 and 𝑦 = 6 𝑥 4 𝑥 + 2 0 have a common tangent at the point of intersection? If so, give the equation of this tangent.

  • AYes, 𝑦 + 4 𝑥 1 4 = 0
  • BYes, 𝑦 4 𝑥 2 2 = 0
  • CNo
  • DYes, 𝑦 8 𝑥 2 6 = 0

Q12:

Determine the equation of the line tangent to the curve 𝑦 = 4 𝑥 2 𝑥 + 4 at point ( 1 , 2 ) .

  • A 𝑦 = 1 4 𝑥 + 1 2
  • B 𝑦 = 8 𝑥 + 6
  • C 𝑦 = 1 6 𝑥 2
  • D 𝑦 = 1 6 𝑥 + 1 4
  • E 𝑦 = 1 6 𝑥 + 1 6

Q13:

The line 𝑦 + 2 𝑥 + 𝑎 = 0 is tangent to the curve 𝑦 = 𝑥 1 at the point ( 𝑏 , 𝑐 ) . Find 𝑎 , 𝑏 , and 𝑐 .

  • A 𝑎 = 4 , 𝑏 = 2 , 𝑐 = 3
  • B 𝑎 = 2 , 𝑏 = 1 , 𝑐 = 0
  • C 𝑎 = 4 , 𝑏 = 2 , 𝑐 = 3
  • D 𝑎 = 2 , 𝑏 = 1 , 𝑐 = 0

Q14:

The line 𝑦 = 5 𝑥 + 4 is tangent to the graph of function 𝑓 at the point ( 1 , 1 ) . What is 𝑓 ( 1 ) ?

Q15:

The line 5 𝑥 + 𝑦 = 2 2 touches the curve 𝑦 = 𝑎 𝑥 + 𝑏 𝑥 4 𝑥 + 2 3 at the point ( 1 , 1 7 ) . Find 𝑎 and 𝑏 .

  • A 𝑎 = 7 , 𝑏 = 5
  • B 𝑎 = 5 , 𝑏 = 3
  • C 𝑎 = 7 , 𝑏 = 5
  • D 𝑎 = 3 , 𝑏 = 5

Q16:

Find the equation of the tangent to the curve 𝑦 = 𝑥 2 𝑥 at the point ( 𝑥 , 3 ) on the curve.

  • A 4 𝑦 𝑥 9 = 0 , 4 𝑦 𝑥 + 1 1 = 0
  • B 𝑦 + 4 𝑥 1 5 = 0 , 𝑦 4 𝑥 7 = 0
  • C 4 𝑦 𝑥 + 1 5 = 0 , 4 𝑦 𝑥 1 3 = 0
  • D 𝑦 4 𝑥 + 9 = 0 , 𝑦 + 4 𝑥 + 1 = 0

Q17:

Find the equation of the tangent to the curve 𝑦 = 8 𝑥 + 5 𝑥 6 at the point ( 1 , 3 ) .

  • A 1 1 𝑦 𝑥 3 4 = 0
  • B 𝑦 1 1 𝑥 8 = 0
  • C 1 1 𝑦 𝑥 + 3 2 = 0
  • D 𝑦 + 1 1 𝑥 + 1 4 = 0

Q18:

List the equations of the normals to 𝑦 = 𝑥 + 2 𝑥 at the points where this curve meets the line 𝑦 4 𝑥 = 0 .

  • A 𝑦 2 𝑥 = 0 , 𝑦 6 𝑥 + 4 = 0
  • B 𝑦 + 2 𝑥 = 0 , 𝑦 + 6 𝑥 4 = 0
  • C 2 𝑦 𝑥 = 0 , 6 𝑦 𝑥 4 6 = 0
  • D 2 𝑦 + 𝑥 = 0 , 6 𝑦 + 𝑥 5 0 = 0

Q19:

Determine the equation of the line normal to the curve 4 𝑦 = 𝑥 at the point ( 2 , 2 ) .

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

Q20:

Determine the equation of the common tangent to the two curves 𝑦 = 5 𝑥 𝑥 4 and 𝑦 = 9 𝑥 + 7 𝑥 .

  • A 1 1 𝑦 + 𝑥 2 1 = 0
  • B 𝑦 1 1 𝑥 1 3 = 0
  • C 1 1 𝑦 𝑥 2 3 = 0
  • D 𝑦 + 1 1 𝑥 + 9 = 0

Q21:

Find the equations to the tangent lines of the curve 𝑦 = ( 𝑥 + 8 ) ( 𝑥 + 1 0 ) at the points where this curve intersects the 𝑥 -axis.

  • A 𝑦 + 2 𝑥 + 1 6 = 0 , 𝑦 2 𝑥 2 0 = 0
  • B 𝑦 2 𝑥 + 1 6 = 0 , 𝑦 + 2 𝑥 2 0 = 0
  • C 𝑦 + 2 𝑥 1 6 = 0 , 𝑦 2 𝑥 + 2 0 = 0
  • D 𝑦 2 𝑥 1 6 = 0 , 𝑦 + 2 𝑥 + 2 0 = 0

Q22:

Determine the equation of the tangent to the curve 𝑦 = 𝑥 8 𝑥 𝑥 + 3 at the point ( 1 , 1 8 ) .

  • A 𝑦 + 8 4 𝑥 + 6 6 = 0
  • B 𝑦 + 4 7 𝑥 + 2 9 = 0
  • C 𝑦 + 7 𝑥 2 5 = 0
  • D 𝑦 7 𝑥 2 5 = 0

Q23:

Find the equation of the tangent to the curve 𝑦 = 5 𝑥 + 8 𝑥 + 5 𝑥 + 6 at the point with 𝑥 -coordinate 0.

  • A 𝑥 = 0
  • B 5 𝑦 + 𝑥 3 0 = 0
  • C 𝑦 6 = 0
  • D 𝑦 5 𝑥 6 = 0

Q24:

Find the equation of the tangent to the curve 𝑦 = 6 𝑥 t a n at 𝑥 = 𝜋 4 .

  • A 7 𝑦 + 8 4 𝑥 2 4 = 0
  • B 4 2 𝑥 7 𝑦 2 4 = 0
  • C 4 2 𝑥 1 4 𝑦 1 8 = 0
  • D 7 𝑦 8 4 𝑥 + 2 4 = 0

Q25:

Find the equation of the normal to the curve 𝑦 = 𝑥 t a n at 𝑥 = 𝜋 4 .

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

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