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Differentiation and Its Applications Questions and Answers
Find the coordinates of the turning points of the curve whose #y=6+2x-4x^2#
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1.
The equation of a curve is given by #y= x^3 – 4x^2 – 3x# (a) Find the value of y when x = -1. (1 mark) (b) Determine the stationary points of the curve. (5 marks) (c) Find the equation of the normal to the curve at x = 1
2.
The gradient of the curve #y = 2x^3 – 9x^2 + px -1# at x =4 is 36. (a) Find: (i) The value of p; (3 marks) (ii) The equation of the tangent to the curve at x = 0.5. (b) Find the co-ordinates of the turning points of the curve.
3.
The gradient of the tangent to the curve # y = ax^3 + bx # at the point (1,1) is -5. Calculate the values of a and b.
4.
The equation of a curve is # y = 2x^3 + 3x^2 #. (a) Find: (i) The x-intercept of the curve: (ii) The y-intercept of the curve. (1 mark) (b) (i) Determine the stationary points of the curve. (ii) For each point in (b) (i) above, determine whether it is a maximum or a minimum. (c) Sketch the curve.
5.
The equation of a curve is given as #y = 2x^3 - (9)/(2)x^2 - 15x + 3# (a) Find: (i) The value of y when x = 2; (2 marks) (ii) The equation of the tangent to the curve at x = 2. (b) Determine the turning points of the curve.
6.
The equation of a curve is given as #y = (1)/(3)x^3 – 4x + 5#. Determine: (a) The value of y when x = 3; (b) The gradient of the curve at x = 3; (c) The turning points of the curve and their nature.
7.
Find the coordinates of the turning points of the curve whose #y=6+2x-4x^2#
8.
a).i).Find the coordinates of the stationary points on the curve #y =x^3 – 3x +2#. ii). For each stationary point determine whether it is minimum or maximum. b).Sketch the graph of the function #y = x^3 -3x +2#.
9.
The velocity V m/s, of a moving body at time t seconds is given by #V = 5t^2– 12t +7#.Calculate the acceleration when t = 2 seconds.
10.
A stone is thrown vertically upwards from a point O. After t seconds, the stone is S metres from O. Given that #S= 29.4t-4.9t^2#,find the maximum height reached by the stone.
11.
A curve is represented by the function #y =(1)/(3)x^3 + x^2 – 3x +2.# a). Find #(dy)/(dx)# b).Determine the values of y at the turning points of the curve. c).Sketch the curve #y = (1)/(3)x^3 + x^2 – 3x + 2#
12.
A particle moves along a straight line such that its displacement s metres from a given point is #S=t^3 -5t^2 +3t +4# where t is time in seconds. Find: a).The displacement of the particle at t =5 seconds. b).The velocity of the particle when t =5 seconds. c) The values of t when the particle is momentarily at rest. d).The acceleration of the particle when t=2 seconds.
13.
The sum of two numbers x and y is 40.Write down the expression, in terms of x for the sum of the squares of the two numbers. Hence determine the minimum value of #x^2 +y^2#
14.
The distance s metres from a fixed point O, covered by a particle after t seconds is given by the equation #s= t^3 -6t^2 + 9t +5#. a).Calculate the gradient of the curve at t=0.5 seconds. b).Determine the values of s at the maximum and the minimum turning points of curves. c).Sketch the curve of #s= t^3 -6t^2 +9t+5#.
15.
A rectangular box open at the top has a square base. The internal side of the base is x cm long and the total internal surface area of the box is #432 cm^2# a) Express in terms of x i).The internal height, h of the box. ii).The internal volume, V of the box. b).Find: i).The value of x for which the internal volume V box is maximum. ii).The maximum internal volume of the box.
16.
The displacement,s metres,of a moving particle after t secs is given by #S =2t^3 -5t^2 + 4t +2# Determine : a)The velocity of the particle when t=3 seconds b)The value of t when the particle is momentarily at rest. c).The displacement when the particle is momentarily at rest. d).The acceleration of the particle when t=3 seconds.
17.
The acceleration of a body moving along a straight line is #(4-t) ms^2# and its velocity is v m/s after t seconds. a).i) If the initial velocity of the body is 3m/s, express the velocity v in terms of t. ii).Find the velocity of the body after 2 seconds. b) Calculate: i).The time taken to attain maximum velocity. ii).The distance covered by the body to attain the maximum velocity.
18.
The displacement of, s metres, of a moving particle from the point O, after t seconds is given by, #s = t^3 – 5t^2 + 3t + 10# a) Find s when t= 2 b) Determine: i)The velocity of the particle when t=5. ii)The value of t when the particle is momentarily at rest. c).Find the time, when the velocity of the particle is maximum.
19.
Given the curve #y= 2x^3 + (1)/(2) x^2 – 4x + 1#. , find the: i).Gradient of the curve at #(1,-frac(1)(2))# ii).Equation of the tangent to the curve at #(1,-frac(1)(2) )#
20.
The displacement s metres of a particle moving along a straight line after t seconds is given by #s = 3t + (3)/(2)t^2 – 2t^3# a).Find the initial acceleration. b).Calculate: i).The time when the particle was momentarily at rest. ii). Its displacement by the time it comes to rest momentarily. c).Calculate the maximum speed attained.
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