Advertisements
Advertisements
Question
Find the tangent and normal to the following curves at the given points on the curve
y = x4 + 2ex at (0, 2)
Advertisements
Solution
y = x4 + 2ex at (0, 2)
Differentiating w.r.t. ‘x’
`("d"y)/("d"x)` = 4x3 + 2ex
Slope of the tangent ‘m’
`(("d"y)/("d"x))_(((0, 2))` = 4(0)3 + 2e0 = 2
Slope of the Normal `- 1/"m" = - 1/2`
Equation of tangent is
y – y1 = m(x – x1)
⇒ y – 2 = 2(x – 0)
⇒ y – 2 = 2x
⇒ 2x – y + 2 = 0
Equation of Normal is
y – y1 = `- 1/"m"` (x – x1)
y – 2 = `- 1/2` (x – 0)
2y – 4 = – x
x + 2y – 4 = 0
APPEARS IN
RELATED QUESTIONS
A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t2 + 3t metres. Find the instantaneous velocities at t = 3 and t = 6 seconds
A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s = 16t2 in t seconds. How long does the camera fall before it hits the ground?
If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = `sqrt(3x)` then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres
A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water?
A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shoreline. How fast is the beam moving along the shoreline when it makes an angle of 45° with the shore?
A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall. How fast is the top of the ladder moving down the wall?
Find the slope of the tangent to the following curves at the respective given points.
y = x4 + 2x2 – x at x = 1
Find the points on the curve y2 – 4xy = x2 + 5 for which the tangent is horizontal
Find the equations of the tangents to the curve y = 1 + x3 for which the tangent is orthogonal with the line x + 12y = 12
Find the equations of the tangents to the curve y = `- (x + 1)/(x - 1)` which are parallel to the line x + 2y = 6
Find the angle between the rectangular hyperbola xy = 2 and the parabola x2 + 4y = 0
Show that the two curves x2 – y2 = r2 and xy = c2 where c, r are constants, cut orthogonally
Choose the correct alternative:
The position of a particle moving along a horizontal line of any time t is given by s(t) = 3t2 – 2t – 8. The time at which the particle is at rest is
Choose the correct alternative:
Find the point on the curve 6y = x3 + 2 at which y-coordinate changes 8 times as fast as x-coordinate is
Choose the correct alternative:
The slope of the line normal to the curve f(x) = 2 cos 4x at x = `pi/12` is
Choose the correct alternative:
The maximum slope of the tangent to the curve y = ex sin x, x ∈ [0, 2π] is at
