Write the equation of the tangent line and the normal to the curve y = x³ – 3x + 2 at the point (2, 4)

a) Tangent is 9x + y = 14; Normal is x – 9y = 38

b) Tangent is 9x − y = 14; Normal is x + 9y = 38

c) Tangent is x + 8y = 24; Normal is 8x – y = 52

d) Tangent is 8x − y = 24; Normal is x + 8y = 52

Write the equation of the tangent line and the normal to the parabola y = 2x² – x + 5 at x = − 0.5

a) Tangent is 6x − 2y = 9; Normal is 2x + 6y = 37

b) Tangent is 6x + 2y = 9; Normal is 2x − 6y + 37 = 0

c) Tangent is 2x − 6y = 9; Normal is 6x + 2y = 37

d) Tangent is 2x + 6y = 9; Normal is 6x − 2y + 37 = 0

Write the equation of the tangent line and the normal to the curve y = x⁴ + 3x² – 16 at the point of intersection with the parabola y = 3x²

a) Tangent is 44x − y = 76; Normal is 44y + x = 530

b) Tangent is 44x + y + 76 = 0; Normal is x − 44y + 530 = 0

c) Both (a) and (b)

d) None of the above

Find the points on the curve y = x³ – 3x + 5 at which the tangent line is parallel to the straight line y = − 2x

a)

b)

c)

d)

Find the points on the curve y = x³ – 3x + 5 at which the tangent line is perpendicular to the straight line  

a) (−2, 7), (2, 3)

b) (2, 7) (−2, 3)

c) (2, 3) (−2, 7)

d) (−2, −3)(−2, −7)

Find the angles at which the straight line y = 4 – x intersects
the parabola

a)

b)

c) Both (a) and (b)

d) Neither of the above

Find the angles at which the sinusoid y = sin x and the cosine curve y = cos x intersect.

a)

b)

c)

d)

Prove that the segment of the tangent to the hyperbola  which is intercepted between the coordinate axes is bisected at the point of tangency.

Find the slope of the tangent to the curve x = t² + 3t − 8, y = 2t² – 2t – 5 at the point M (2, − 1)

a)

b)

c)

d)

Find the equations of the tangent and the normal to the following curve
4x³ – 3xy² + 6x² – 5xy – 8y² + 9x + 14 = 0
at the point (− 2, 3)

a) Tangent is 9x + 2y + 12 = 0; Normal is 2x – 9y + 31 = 0

b) Tangent is 9x − 2y + 12 = 0; Normal is 2x + 9y + 31 = 0

c) Tangent is 2x + 9y + 12 = 0; normal is 9x – 2y + 31 = 0

d) Tangent is 2x − 9y + 12 = 0; Normal is 9x + 2y + 31 = 0

Find the equations of the tangent and the normal to the following curve
x⁵ + y⁵ – 2xy = 0 at the point (1, 1)

a) Tangent is x + y = 1; Normal is x – y = 2

b) Tangent is x + y = 2; Normal is x = y

c) Tangent is x = y; Normal is x + y = 1

d) Tangent is x = y + 1; Normal is x + y = 2

f (x) = 3x⁵ – 15x³ + 5x – 7
Find out at which of the points x the rate of change of the function is minimal.

a)

b)

c)

d) None of the above

A point is moving along a cubic parabola 12y = x³. When does its ordinate change faster than its abscissa?

a) – 2 < x < 2

b) x > 2

c) -2 < x < 0

d) (b) or (c)

A raft is pulled to the bank by means of a rope which is wound on a drum, at a rate of 3 m/min. Determine the speed of the raft at the moment when it is 25m distant from the bank if the drum is situated on the bank 4m above water level.

a) 25.3m/min

b) 3.03m/min

c) 25m/min

d) 28m/min

Find the inclination with the X-axis of the tangent to the cubic parabola y = x³ at the point

a) 0

b)

c)

d)

Write the equations of the tangents to the curve  at the points of its intersection with the hyperbola

a) y = 1, x + 2y = 2

b) y + 1 = 0, 2x + y = 1

c) y + 1 = 0, 2x – y + 2 = 0

d) y = 1, x + 2y + 1 = 0

The velocity of a body in rectilinear motion is determined by the formula v = 3t + t². What acceleration will the body have 4 seconds after the start?

a) 3

b) 8

c) 11

d) 14

The law of rectilinear motion of a body with mass 100kg is s = 2t² + 3t + 1 meters. Find the kinetic energy (mv²/2) of the body 5 seconds after the start.

a) 23

b) 529

c) 5290000

d) 26450000

Show that if the law of motion of a body is s = a + b, then its acceleration is numerically equal to the distance covered.

By differentiating the formula
cos 3x = cos³ x – 3 cos x sin² x
deduce the formula
sin 3x = 3 cos² x sin x – sin³ x

From the formula for the sum of the geometric progression
 

Deduce the formula for the following sum
1 + 2x + 3x² + .  .  .+nx^{n – 1}

Find y’ if
y = f (sin²x) + f (cos²x)

Find y’ if