📖 Hover (or tap) any dotted word for its definition. Write your answers in the squared boxes with a pencil, finger or mouse — they save on your device.
(a)A function is defined for x ∈ ℝ by:
f(x) = 6 + x2 + sin 4x
(i)Find f′(x), the derivative of f with respect to x.
(a)(i)
(ii)Find the equation of the tangent to the curve y = f(x) at the point where x = 0. Give your answer in the form ax + by + c = 0, where a, b, c ∈ ℤ.
(a)(ii)
(a)The function f(x) is defined for x ∈ ℝ by:
f(x) = (3x5 − 4)28
Find an expression for f′(x) (use the chain rule). You do not need to simplify your answer.
(a)
(b)The function g(x) is defined for x ∈ ℝ, x ≠ 3·5, by:
g(x) = 32x − 7
By finding g′(x), show that this function has no local maximum or minimum points.
(b)
Jacob takes part in a race, which involves kayaking and running. He must get from the point S in the sea to the point F on the coastline. A is another point on the coastline. |SA| = 2 km, |AF| = 8 km, and ∠SAF is a right angle. Jacob kayaks at an average speed of 6 km/hour and runs at an average speed of 12 km/hour.
(c)Find how long it would take Jacob in total to kayak from S to A, and then run from A to F.
(c)
(d)Find how long it would take Jacob to kayak directly from S to F, correct to the nearest minute.
(d)
The point B is on [AF], and |AB| = x km, where 0 ≤ x ≤ 8, x ∈ ℝ.
(e)Jacob is going to kayak from S to B, and then run from B to F.
(i)Find an expression in x for T, the total time, in hours, that it will take Jacob to get from S to B to F.
(e)(i)
(ii)Jacob knows that there is one value of x for which T(x) is a minimum (0 ≤ x ≤ 8). Solve the following equation to find this value of x, correct to 3 decimal places:
T′(x) = x6√(x2 + 4) − 112 = 0
(e)(ii)
(iii)There are no other values of x in [0, 8] for which T′(x) = 0, apart from the answer to part (e)(ii). Use this fact, and your answers from parts (c) and (d), to find which one of the following graphs could represent T(x), for x ∈ [0, 8]. Justify your answer fully.
(e)(iii) — answer & justification
Dani drives a car. The fuel consumption, F, of Dani’s car depends on the speed of the car, c. For one particular journey, F is given by F(c) = 0·05c2 − 8·5c + 800, where F is in litres per 10 000 km, and c is in km/hour, with 40 ≤ c ≤ 120.
(i)Show that there is no difference between the fuel consumption (F) when Dani’s car is travelling at 60 km/hour and when it is travelling at 110 km/hour.
(a)(i)
(ii)Find an expression for dFdc, the rate of change of fuel consumption with respect to speed (its derivative).
(a)(ii)
During part of the journey, the speed, c, of Dani’s car at time t is given by c = 78 + 9 ln(t2), where t is the time in minutes, 1 ≤ t ≤ 10, and c is in km/hour.
(iii)Use this, and your answer to part (a)(ii), to find the value of dFdc when t = 7. Give your answer correct to 1 decimal place.
(a)(iii)
(iv)Show that the rate of change of the car’s speed with respect to time is given by dcdt = 18t.
(a)(iv)
(v)Use your answers to parts (a)(iii) and (a)(iv) to find the rate of change of the car’s fuel consumption, F, with respect to time, at the instant when t = 7 minutes. Give your answer in (litres per 10 000 km) per minute.
(a)(v)
(b)The function f is defined for x ∈ ℝ as: f(x) = 2x3 − 9x2 + 5x − 11
(i)Find the equation of the tangent to the graph of f at the point where x = 2. You do not need to simplify your answer.
(b)(i)
(ii)Find the x co-ordinate of the point of inflection of f.
(b)(ii)
(c)The diagram below shows the curve y = p(x) and the line y = l(x), for 0 ≤ x ≤ 10, x ∈ ℝ.
There are two values of x in the domain 0 ≤ x ≤ 10 for which p′(x) = l′(x), where p′(x) is the derivative of p(x). Use the information in the diagram to estimate these two values of x, as accurately as you can. Show your work on the diagram.
(c)
(a)Differentiate the following function from first principles, with respect to x: f(x) = x2 − 7x − 10
(a)
(b)The function g(x) is defined for x ∈ ℝ by: g(x) = 6x + 1x4 + 3 Find the value of g′(−2), the derivative of g(x) when x = −2.
(b)
(c)h : ℝ → ℝ is a continuous function. The graph of h(x) has a local minimum at the point (0, 5). State whether the following statement is true or false: "The value of h(x) must be at least 5 for all real values of x." Justify your answer.
(c)
(c)A sequence of functions F0, F1, F2, ... is defined as follows, for x ∈ ℝ, x > 0: F0 = x2024 For n ≥ 1, the function Fn is the derivative of Fn−1 with respect to x.
(i)Write F1 and F2 in terms of x.
(c)(i)
(ii)Find the first value of n for which Fn = 0.
(c)(ii)
A hemisphere has a diameter of x cm. Its volume in cm3 is V(x) = π12 x3.
(d)The volume of the hemisphere is increasing at a constant rate of 450 cm3 per second. Find the rate at which the diameter (x) of the hemisphere is increasing with respect to time, when x = 20 cm. Give your answer in cm per second, correct to 1 decimal place. (Remember that V(x) = π12 x3.)
(d)
A company grows and sells plants.
(a)The function W(x) below can be used to model the height, in mm, of a water spinach plant, for the first 35 days after it starts to grow: W(x) = 0·667x + 1·5x2 − 0·025x3 Here, x is the number of days after the plant starts to grow, where 0 ≤ x ≤ 35, x ∈ ℝ.
(i)Use W(x) to estimate the height of a water spinach plant after 15 days. Give your answer correct to the nearest mm.
(a)(i)
(ii)Write down W′(x), the derivative of W(x).
(a)(ii)
(b)The height of a different plant can be modelled by the function P(x), where x is again the number of days after the plant starts to grow. The derivative of this function is P′(x) = 1·1 + 2·73x − 0·078x2 Find the range of values of x for which P′(x) > 24. In your answer, give each value correct to the nearest whole number.
(b)
(a)f(x) = x2 + bx + c, where b, c ∈ ℝ. f(x) has a local minimum point at (3, −1). Find the value of b and the value of c.
(a)
(a)The function f is defined as follows, for x ∈ ℝ: f(x) = 15x2 + 7 Find f′(x), the derivative of f. Give your answer in its simplest form.
(a)
(b)The function g(x) is defined as follows, for x ∈ ℝ, 0 < x < π: g(x) = (tan(x2))(ln x) Find the value of g′(π2). Give your answer in the form a + ln b, where a, b ∈ ℝ.
(b)
Fiona is driving on a motorway. She passes a point A. Her speed is given by v(t) = 23t3 − 6t2 + 13t + 109, where v is her speed in km/hour t minutes after passing A, for 0 ≤ t ≤ 5 and t ∈ ℝ.
(a)Work out Fiona's speed when she passes the point A.
(a)
(b)Work out Fiona's acceleration (the rate at which her speed is increasing) 5 minutes after she passes the point A. Give your answer in km/hour per minute.
(b)
(c)Find the time (value of t) at which Fiona reaches her maximum speed, during the first 4 minutes after she passes the point A. Give your answer correct to 2 decimal places.
(c)
(e)Taking v′(t) to be the derivative of v, and v″(t) the second derivative: v′(1) > 0 and v″(1) < 0. Close to t = 1, the graph of y = v(t) must look like one of the four graphs below. Write down which graph this is. Justify your answer, using both v′(1) and v″(1).
(e)
There is an Average Speed Zone on the motorway, starting at point A and ending at point B. The distance from A to B along the motorway is 10 km. Cameras record the time taken for each car from A to B, and each car's average speed is calculated.
(f)Work out the minimum time, in minutes, that a driver could get from A to B, while not driving above 100 km/hour.
(f)
(g)Rohan drives from A to B. He passes A at a constant speed of 120 km/hour. After 2 minutes at this speed, he starts to decelerate at a constant rate, until he reaches B. Overall, his average speed from A to B is 100 km/hour. Work out Rohan's deceleration. Give your answer in km/hour per minute.
(g)
(i)A tangent to the curve y = 12x is drawn at the point (p, 12p), where p ∈ ℝ and p > 0. Show that the equation of this tangent is y = −12p2 x + 24p
(c)(i)
(ii)The area of the triangle formed by the x-axis, the y-axis, and the tangent y = −12p2 x + 24p is always k square units, where k ∈ ℕ is a constant. Work out the value of k.
(c)(ii)
A square-based pyramid has base sides c and perpendicular height h. A horizontal square a distance x down from the top has area S(x) = x2 c2h2.
(e)(ii)x starts to increase at a rate of 3 units per second. This causes S(x) to increase as well. Find the rate of change of S(x) with respect to time, at the instant when x is half the perpendicular height of the pyramid. Give your answer in square units per second, in terms of c and h.
(e)(ii)
(a)g(x) = x2 − 1x, where x ∈ ℝ. Find g′(x), the derivative of g(x).
(a)
(b)f(x) = 2x3 − 21x2 + 40x + 63, where x ∈ ℝ.
(ii)Find the range of values of x for which f′(x) is negative, correct to 2 decimal places.
(b)(ii)
(a)Differentiate f(x) = 2x2 + 4x with respect to x, from first principles.
(a)
(b)A rectangle is expanding in area. Its width is x cm, where x ∈ ℝ and x > 0. Its length is always four times its width. Find the rate of change of the area of the rectangle with respect to its width, x, when the area of the rectangle is 225 cm2.
(b)
Hannah is doing a training session. Her heart-rate, h(x), is in beats per minute (BPM), where x is the time in minutes from the start, x ∈ ℝ. For the first 8 minutes, h(x) = 2x3 − 28·5x2 + 105x + 70.
(a)Work out Hannah's heart-rate 4 minutes after the start of the session.
(a)
(b)Find h′(x).
(b)
(c)Find h′(2), and explain what this value means in the context of Hannah's heart-rate.
(c)
(d)Find the least value and the greatest value of h(x), for 0 ≤ x ≤ 8. Use calculus in your solution (the graph above is to scale).
(d)
(e)How long after the start is Hannah's heart-rate decreasing most quickly, within the first 8 minutes? Give your answer in minutes and seconds.
(e)
(f)Bruno, Karen, and Martha start at the same time as Hannah (all heart-rates in BPM).
(i)Bruno's heart-rate, b(x), is always 15 BPM more than Hannah's. Write b′(x) in terms of h′(x), for 0 ≤ x ≤ 8.
(f)(i)
(ii)Karen's heart-rate, k(x), is always 10% less than Hannah's. Write k′(x) in terms of h′(x), for 0 ≤ x ≤ 8.
(f)(ii)
(g)Martha does each exercise for a longer time. For 0 ≤ x ≤ 10, Martha's heart-rate is m(x) = h(0·8x). Use h(x) = 2x3 − 28·5x2 + 105x + 70 to write m(x) in the form m(x) = ax3 + bx2 + cx + d, where a, b, c, d ∈ ℝ.
(g)
(a)The derivative of f(x) = 2x3 + 6x2 − 12x + 3 can be expressed in the form f′(x) = a(x + b)2 + c, where a, b, c ∈ ℤ and x ∈ ℝ.
(i)Find the value of a, the value of b, and the value of c.
(a)(i)
(ii)If g(x) = 36x + 5, find the range of values of x for which f′(x) > g′(x).
(a)(ii)
(b)The diagram shows t, the tangent line to h(x) = 2 sin(2x), where 0 ≤ x ≤ π, at the point where x = π6. A(0, k), where k ∈ ℝ, is the point where t cuts the y-axis. Find the value of k correct to two decimal places.
(b)
The diagram shows the graph of h′(x), the derivative of a cubic function h(x).
(a)Show that h′(x) = −2x2 + 4x + 6.
(a)
(b)Use h′(x) to find the maximum positive value of the slope of a tangent to h(x).
(b)
The function h(x) = 0·001x3 − 0·12x2 + 3·6x + 5 models the height above level ground (in metres) of a section of a rollercoaster track, where x is the horizontal distance from a fixed point.
(b)(i) Find h′(x), the derivative of h(x).
(b)(i)
(ii)Show that this section of track reaches its maximum height above level ground when x = 20.
(b)(ii)
(iii)Find, using calculus, the height above ground (in metres) at the instant the track passes through an inflection point.
(b)(iii)
The temperature of a cooling coffee is T(t) = 75e−0·081t + 20 (°C), t in minutes.
(c)Find, to the nearest °C, the temperature the coffee has reached when T′(t) = −4·05, where T′(t) is the rate at which the coffee is cooling, in °C per minute.
(c)
(d)A sugar cube is put into the coffee and keeps its cube shape as it dissolves. Its volume decreases at the constant rate of 120 cm3/sec. Let x(t) be the side length of the cube at time t. Find the rate of change of x(t) when the volume of the cube reaches 164 cm3.
(d)
The diagram below shows two functions f(x) and g(x). The function f(x) is given by the formula f(x) = x3 + kx2 + 15x + 8, where k ∈ ℤ, and x ∈ ℝ.
(a)Given that f′(3) = −12, show that k = −9, where f′(3) is the derivative of f(x) at x = 3.
(a)
(b)The function g(x) is the line that passes through the two turning points of f(x) = x3 − 9x2 + 15x + 8, as shown. Find the equation of g(x).
(b)
(c)Show that the graph of g(x) contains the point of inflection of f(x).
(c)
(a)Differentiate (3x − 5)(2x + 4) with respect to x from first principles.
(a)
(b)(i)h(x) = 12 ln(2x + 3) + C, where C is a constant. Find h′(x), the derivative of h(x).
(b)(i)
A rectangle is inscribed in a circle of radius 5 units and centre O(0, 0) as shown. Let R(x, y), where x, y ∈ ℝ, be the vertex of the rectangle in the first quadrant. Let θ be the angle between [OR] and the positive x-axis, where 0 ≤ θ ≤ π2.
(a)(i)The point R(x, y) can be written as (a cos θ, b sin θ), where a, b ∈ ℝ. Find the value of a and the value of b.
(a)(i)
(ii)Show that A(θ), the area of the rectangle, measured in square units, can be written as A(θ) = 50 sin 2θ.
(a)(ii)
(iii)Use calculus to show that the rectangle with maximum area is a square.
(a)(iii)
(iv)Find this maximum area.
(a)(iv)
(b)A person who is 2 m tall is walking towards a streetlight of height 5 m at a speed of 1·5 m/s. Find the rate of change, in m/s, at which the length of the person's shadow (x), cast by the streetlight, is changing.
(b)
g(x) = 3x ln x − 9x + 4 ln x − 12.
(c)Evaluate g′(e) correct to 2 decimal places (the derivative of g at x = e).
(c)
Norman windows consist of a rectangle topped by a semi-circle as shown. Let the height of the rectangle be y metres and the radius of the semi-circle be x metres. The perimeter of the window is P.
(a)(i)Write P in terms of x, y, and π.
(a)(i)
(ii)In a particular Norman window the perimeter P = 12 metres. Show that y = 12 − (2 + π)x2 for 0 ≤ x ≤ 122 + π, where x ∈ ℝ.
(a)(ii)
(b)Complete the table of values for y = 12 − (2 + π)x2 (at x = 0 and x = 122 + π), draw the graph of this linear function, and find the slope of the graph correct to 2 decimal places. Interpret this slope in the context of the question.
(b)
(c)(i)The Norman window has a perimeter of 12 metres and y = 12 − (2 + π)x2. Show that the function a(x) = 24x − (π + 4)x22 represents the area of the window, in terms of x and π.
(c)(i)
(ii)Find a′(x).
(c)(ii)
(iii)Find the relationship between x and y when the area of the window in part (c)(i) is at its maximum.
(c)(iii)
(a)Let h(x) = cos(2x), where x ∈ ℝ. A tangent is drawn to the graph of h(x) at the point where x = π3. Find the angle that this tangent makes with the positive sense of the x-axis.
(a)
The graph of the symmetric function f(x) = 1√(2π) e−12x2 is shown.
(a)Find the co-ordinates of A, the point where the graph intersects the y-axis. Give your answer in terms of π.
(a)
(b)The co-ordinates of B are (−1, 1√(2πe)). Find the area of the shaded rectangle in the diagram. Give your answer correct to 3 decimal places.
(b)
(c)Use calculus to show that f(x) is decreasing at C.
(c)
(d)Show that the graph of f(x) has a point of inflection at B.
(d)
(a)Differentiate 13x2 − x + 3 from first principles with respect to x.
(a)
(b)f(x) = ln(3x2 + 2) and g(x) = x + 5, where x ∈ ℝ. Find the value of the derivative of f(g(x)) at x = 14 (using the chain rule). Give your answer correct to 3 decimal places.
(b)
f(x) = 2x3 + 5x2 − 4x − 3, where x ∈ ℝ.
(b)Find the co-ordinates of the local maximum point and the local minimum point of the function f.
(b)
(c)f(x) + a, where a is a constant, has only one real root. Find the range of possible values of a.
(c)
(a)Differentiate the function (2x + 4)2 from first principles, with respect to x.
(a)
(b)(i)If y = x sin(1x), find dydx (using the product rule and chain rule).
(b)(i)
(ii)Find the slope of the tangent to the curve y = x sin(1x), when x = 4π. Give your answer correct to two decimal places.
(b)(ii)
(a)(i)Air is pumped into a spherical exercise ball at the rate of 250 cm³ per second. Find the rate of change at which the radius is increasing when the radius is 20 cm. Give your answer in terms of π (related rates).
(a)(i)
(ii)Find the rate at which the surface area of the ball is increasing when the radius is 20 cm.
(a)(ii)
Sarah's first throw at the basket: the ball left her hands at A and entered the basket at B. With A(−0·5, 2·565) and B(4·5, 3·05), the path of the centre of the ball is f(x) = −0·274x2 + 1·193x + 3·23, with x and f(x) in metres.
(a)(i)Find the maximum height reached by the centre of the ball, correct to three decimal places.
(a)(i)
(ii)Find the acute angle to the horizontal at which the ball entered the basket. Give your answer correct to the nearest degree.
(a)(ii)
(iii)Sarah's second throw followed parabola g(x), the image of f(x) under the translation mapping A onto C(0, 2). Using your result from part (a)(i), show that the centre reached its maximum height at (2·677, 3·964), correct to three decimal places.
(a)(iii)
(iv)Hence, or otherwise, find the equation of the parabola g(x).
(a)(iv)
(b)Differentiate x − √(x + 6) with respect to x.
(b)
(c)Find the co-ordinates of the turning point of the function y = x − √(x + 6), x ≥ −6.
(c)
A plane flying horizontally at P (150 m above level ground) begins its descent. P is 5 km horizontally from touchdown O, and the plane lands horizontally at O. Taking O as the origin, (x, f(x)) describes the descent, where f(x) = 0·0024x3 + 0·018x2 + cx + d, −5 ≤ x ≤ 0, in km.
(a)(i)Show that d = 0.
(a)(i)
(ii)Using the fact that P is the point (−5, 0·15), or otherwise, show that c = 0.
(a)(ii)
(b)(i)Find the value of f′(x), the derivative of f(x), when x = −4.
(b)(i)
(ii)Use your answer to part (b)(i) to find the angle at which the plane is descending when it is 4 km from touchdown. Give your answer correct to the nearest degree.
(b)(ii)
(c)Show that (−2·5, 0·075) is the point of inflection of the curve y = f(x).
(c)
(d)(i)If (x, y) is a point on the curve y = f(x), verify that (−x − 5, −y + 0·15) is also a point on y = f(x).
(d)(i)
(ii)Find the image of (−x − 5, −y + 0·15) under symmetry in the point of inflection.
(d)(ii)
An oil-spill occurs off-shore in calm water. The oil spills at a rate of 4 × 106 cm³ per minute and floats on top of the water.
(a)(i)Complete the table showing the total volume of oil on the water after each of the first 6 minutes.
(a)(i)
(ii)Draw a graph to show the total volume of oil on the water over the first 6 minutes.
(a)(ii)
(iii)Write an equation for V(t), the volume of oil on the water, in cm³, after t minutes.
(a)(iii)
(b)The spilled oil forms a circular slick 1 millimetre thick.
(i)Write an equation for the volume of oil in the slick, in cm³, when the radius is r cm.
(b)(i)
(ii)Find the rate of change at which the radius of the oil slick is increasing when the radius is 50 m (related rates).
(b)(ii)
(c)Show that the area of water covered by the oil slick is increasing at a constant rate of 4 × 107 cm² per minute.
(c)
(d)The nearest land is 1 km from the point at which the oil-spill began. Find how long it will take for the oil slick to reach land. Give your answer correct to the nearest hour.
(d)
(a)Differentiate the function 2x2 − 3x − 6 with respect to x from first principles.
(a)
(b)Let f(x) = 2xx − 2, x ≠ 2, x ∈ ℝ. Find the co-ordinates of the points at which the slope of the tangent to the curve y = f(x) is 14.
(b)
ADEC is a rectangle with |AC| = 7 m and |AD| = 2 m, as shown. B is a point on [AC] such that |AB| = 5 m. P is a point on [DE] such that |DP| = x m.
(i)Let f(x) = |PA|2 + |PB|2 + |PC|2. Show that f(x) = 3x2 − 24x + 86, for 0 ≤ x ≤ 7, x ∈ ℝ.
(i)
(ii)The function f(x) has a minimum value at x = k. Find the value of k and the minimum value of f(x).
(ii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.