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(c)k ∈ ℝ is a constant, and:
∫0 k e5x dx = 9
Use this (an example of integration) to find the value of k. Write your answer in the form k = ln ab, where a, b ∈ ℕ.
(c)
(b)Over the first 8 seconds that Dani is driving her car, the car’s speed, in km/hour, can be approximated using the following function v(t):
v(t) = 8e0·4t − 8, for 0 ≤ t ≤ 4
v(t) = −t2 + 24t − 48·4, for 4 < t ≤ 8
v(t) = −t2 + 24t − 48·4, for 4 < t ≤ 8
where t ∈ ℝ is the time, in seconds, after the car starts moving.
(i)Fill in the table below to show the values of v(t) for the given values of t, up to t = 8. Give each value correct to 1 decimal place, where appropriate.
| Time, t (s) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| Speed, v (km/h) | 0 | 9·8 | 31·6 | 70·6 | 79·6 |
(b)(i) working
(ii)Hence, draw the graph of the function y = v(t) on the axes below, for 0 ≤ t ≤ 8, t ∈ ℝ.
Plot the values and draw y = v(t) directly on the axes
(iii)Use integration, and v(t) = −t2 + 24t − 48·4, to find the average speed of Dani’s car for 4 < t ≤ 8. Give your answer in km/hour, correct to 1 decimal place.
(b)(iii)
(a)Find the integral: ∫ cos 6x dx
(a)
(c)The logo for the company is shown on the co-ordinate diagram below. The logo is the region enclosed by three curves, defined by the following functions c, s, and k: c(x) = x2, for −1 ≤ x ≤ 1
s(x) = 2x − x2, for 0 ≤ x ≤ 1 k(x) is the image of s(x) under axial symmetry in the y-axis.
s(x) = 2x − x2, for 0 ≤ x ≤ 1 k(x) is the image of s(x) under axial symmetry in the y-axis.
(i)Use integration to work out the area of the logo. Hint: find the area of the logo in the first quadrant, and then double it.
(c)(i)
(ii)The function k can be written in the form k(x) = −x2 + bx + c, where b, c ∈ ℝ are constants. Find the value of b and the value of c. Remember that k is the image of s under axial symmetry in the y-axis.
(c)(ii)
(a)f and g are two functions of x ∈ ℝ, where f(x) = x + 4 and g(x) = x2 − 2.
(i)Find the two values of x for which f(x) = g(x).
(a)(i)
(ii)Find the area of the shaded region in the diagram below (not to scale), the region between the graphs of f(x) = x + 4 and g(x) = x2 − 2, from x = −1 to x = 2.
(a)(ii)
(b)b ∈ ℝ is a positive constant, and: ∫0b b ebx dx = e Work out the value of b.
(b)
Fiona's speed t minutes after passing point A is v(t) = 23t3 − 6t2 + 13t + 109 km/hour, for 0 ≤ t ≤ 5.
(d)Use integration to work out Fiona's average speed over the 5 minutes after she passes the point A. Give your answer correct to 2 decimal places.
(d)
(e)Diagram A shows a square-based pyramid, with base sides of length c units, horizontal base, and perpendicular height h units (c, h ∈ ℝ). Diagram B shows the same pyramid with a horizontal square that lies within it, a distance of x units down from the top, where 0 < x < h. The area of the shaded square is S(x) = x2 c2h2.
(i)The volume of the pyramid is ∫0h S(x) dx. Use integration to find the volume of the pyramid in cubic units, in terms of c and h.
(e)(i)
(a)g(x) = 2x2 + 5x + 6, where x ∈ ℝ. Find ∫ g(x) dx.
(a)
(b)The diagram shows the graph of f(x) = ax2 + bx + c, where a, b, c ∈ ℤ. Three regions K, L, and N are each bounded by the x-axis, the graph of f(x), and two vertical lines.
(i)The area of region K is 538 square units. Use integration of f(x) to show that 4a + 3b + 3c = 807.
(b)(i)
(ii)The areas of K, L, and N give three equations: 4a + 3b + 3c = 807
28a + 9b + 3c = 879
76a + 15b + 3c = 663 Solve these to find the values of a, b, and c.
28a + 9b + 3c = 879
76a + 15b + 3c = 663 Solve these to find the values of a, b, and c.
(b)(ii)
A point A on a Ferris wheel has height h(t) = 72 − 60 cos(π3 t) metres after t minutes.
(f)Use integration to find the average height of the point A over the first 8 minutes that the wheel is turning. Give your answer correct to 1 decimal place.
(f)
A cubic function h(x) has derivative h′(x) = −2x2 + 4x + 6.
(c)The graph of h(x) passes through the point (0, −2). Find the equation of h(x).
(c)
(c)Use the function h(x) = 0·001x3 − 0·12x2 + 3·6x + 5, x ∈ ℝ, to find the average height of this section of the track above level ground, from x = 0 to x = 75. Give your answer in metres correct to 2 decimal places.
(c)
The function h(x) = 12 ln(2x + 3) + C has derivative h′(x). The diagram below shows part of the graph of the function h′(x). The shaded region is between the graph and the x-axis, from x = 0 to x = A.
(b)(ii)This shaded region has an area of ln 3 square units. Find the value of A (using the definite integral of h′(x)).
(b)(ii)
(a)Find ∫(4x3 − 6x + 10) dx (the indefinite integral).
(a)
(b)Part of the graph of a cubic function f(x) is shown (graph not to scale). The graph cuts the x-axis at the three points A(2, 0), B, and C.
(i)Given that f′(x) = 6x2 − 54x + 109, show that f(x) = 2x3 − 27x2 + 109x − 126 (using the anti-derivative and the point A).
(b)(i)
(ii)Find the co-ordinates of the point B and the point C.
(b)(ii)
h(x) = cos(2x), where x ∈ ℝ.
(b)Find the average value of h(x) over the interval 0 ≤ x ≤ π4, x ∈ ℝ. Give your answer in terms of π.
(b)
Parts of the graphs of the functions h(x) = x and k(x) = x3, x ∈ ℝ, are shown.
(a)Find the co-ordinates of the points of intersection of the graphs of the two functions.
(a)
(b)(i)Find the total area enclosed between the graphs of the two functions.
(b)(i)
(ii)On the diagram, using symmetry or otherwise, draw the graph of k−1, the inverse function of k.
(b)(ii)
The graph of the function g(x) = ex, x ∈ ℝ, 0 ≤ x ≤ 1, is shown.
(a)On the same diagram, draw the graph of h(x) = e−x, x ∈ ℝ, in the domain 0 ≤ x ≤ 1.
(a)
(b)Find the area enclosed by g(x) = ex, h(x) = e−x, and the line x = 0·75. Give your answer correct to 4 decimal places.
(b)
The inflated ball is kicked into the air from a point O on the ground. Taking O as the origin, (x, f(x)) describes the path of the ball, where f(x) = −x2 + 10x and both x and f(x) are in metres.
(b)(i)Find the values of x when the ball is on the ground.
(b)(i)
(ii)Find the average value (average height) of the ball above the ground, during the interval from when it is kicked until it hits the ground again.
(b)(ii)
Let f(x) = −x2 + 12x − 27, x ∈ ℝ.
(a)(i)Complete Table 1 of f(x) values (at x = 3, 4, 5, 6, 7, 8, 9).
(a)(i)
(ii)Use Table 1 and the trapezoidal rule to find the approximate area of the region bounded by the graph of f and the x-axis.
(a)(ii)
(b)(i)Find ∫39 f(x) dx (the exact definite integral).
(b)(i)
(ii)Use your answers above to find the percentage error in your approximation of the area, correct to one decimal place.
(b)(ii)
(a)Find ∫ 5 cos 3x dx.
(a)
(b)The slope of the tangent to a curve y = f(x) at each point (x, y) is 2x − 2. The curve cuts the x-axis at (2, 0).
(i)Find the equation of f(x).
(b)(i)
(ii)Find the average value of f over the interval 0 ≤ x ≤ 3, x ∈ ℝ.
(b)(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.