blog.bernatchez.net - calculus

IB Calculus Problem 1

2026-05-19T02:21:52+00:00

Let the function \(f(x) = 6x^2-3x\) be represented below.

The figure is not to scale.

Find \(\int \! (6x^2-3x) \, \mathrm{d}x\).
Find the area of the region bounded by the graph of \(f(x)\),

the x-axis and the lines \(x = 1\) and \(x = 2\).

That is, \(\int_1^2 \! (6x^2-3x) \, \mathrm{d}x\).

IB Calculus Problem 2

2026-05-19T02:21:50+00:00

A closed cylindrical metal box has a radius of \(r\) centimetres and a height of \(h\) centimetres, with a volume of \(20\pi\, cm^3\).

The figure is not to scale.

Express \(h\) in terms of \(r\).

The metal for the base and lid of the box costs 10 cents per \(cm^2\).

The metal for the curved side costs \(8\) cents per \(cm^2\).

The total cost of the metal, in cents, is \(C\).

Show that \(C\,=\,20\pi{}r^2 + \frac{320\pi}{r}\)
Given that a minimum value of \(C\) exists, find this minimum value in terms of \(\pi\).

IB Calculus Problem 3

2026-05-19T02:21:48+00:00

Consider a function \(f(x)\). The line \(L_1\) with equation \(y = 3x + 1\) is tangent to the graph of \(f(x)\) at \(x = 2\).

1. Write down \(f^\prime(2)\).
2. Find \(f(2)\).

Let \(g(x) = f(x^2 + 1)\) and let \(P\) be the point on the graph of \(g\) where \(x = 1\).

Show that the graph of \(g\) has a slope of 6 at point \(P\).

Let \(L_2\) be the tangent to the graph of \(g\) at point \(P\). \(L_1\) intersects \(L_2\) at point \(Q\).

Find the y-coordinate of \(Q\).

IB Calculus Problem 4

2026-05-19T02:21:46+00:00

The following figure shows the graph of \(f(x) = a\,Cos\,bx\), for \(0 \leq x \leq 4\).

The figure is not to scale.

There is a minimum point at \(P( 2, -3 )\) and a maximum point at \(Q( 4, 3 )\).

1. State the value of \(a\).
2. Find the value of \(b\).
State the gradient of the curve at \(P\).
State the equation of the normal to the curve at \(P\).

IB Calculus Problem 5

2026-05-19T02:21:45+00:00

Let \(h(x) = \frac{6x}{cos x}\).

Find \(h^\prime(0)\).

IB Calculus Problem 6

2026-05-19T02:21:44+00:00

The following figure shows part of the graph of \(f(x) = 2x\sqrt[2]{a^2 - x^2}\), for \(-1 \leq x \leq a\), where \(a > 1\).

The figure is not to scale.

The line \(L\) is the tangent to the graph of \(f\) at the origin, \(O\). The point \(P(a; b)\) is on \(L\).

Given that \(f^\prime(x) =\frac{2a^2 - 4x^2}{\sqrt{a^2-x^2}}\), for \(-1 \leq x \leq a\), find the equation of \(L\).
Hence or otherwise, find an expression for \(b\) in terms of \(a\).

IB Calculus Problem 7

2026-05-19T02:20:02+00:00

The following figure shows part of the graph of the function \(f(x) = 2x^2\).

The figure is not to scale.

The line \(T\) is the tangent to the graph of \(f\) at \(x = 1\).

Show that the equation of \(T\) is \(y = 4x - 2\).
Find the x-intercept of \(T\).
The shaded region \(R\) is bounded by the graph of \(f\), the line \(T\) and the x-axis.
1. Write down an expression for the area of \(R\).
2. Find the area of \(R\).

IB Calculus Problem 8

2026-05-19T02:18:27+00:00

The following figure shows part of the graph of the quadratic function \(f\).

The figure is not to scale.

The x-intercepts are at \(( -4; 0 )\) and \(( 6; 0 )\) and the y-intercept is at \(( 0; 240 )\).

Write down \(f(x)\) in the form \(f(x) = -10(x - p) (x - q)\).
Find another expression for \(f(x)\) in the form \(f(x) = -10(x - h)^2 + k\).
Show that \(f(x)\) can also be written as \(f(x) = 240 + 20x -10x^2\).
A particle moves in a straight line such that its velocity \(v\) ( in \(ms^{-1}\) ), at time \(t\) (in seconds), is given by \(v = 240 + 20t -10t^2\) , for \(0 \leq t \leq 6\).
1. Find the value of \(t\) when the velocity of the particle is greatest.
2. Find the acceleration of the particle when its velocity is zero.

IB Calculus Problem 9

2026-05-19T02:17:12+00:00

Let \(f(x) = kx^4\). The point \(P(1 ; k)\) is on the curve of \(f\). At \(P\), the normal to the curve is parallel to \(y = -\frac{1}{8}x\).

Find the value of \(k\).

IB Calculus Problem 10

2026-05-19T02:14:08+00:00

A function \(f\) is defined for \(-4 \leq x \leq 3\).

The graph of \(f\) is shown below.

The figure is not to scale.

The graph has a local maximum at \(x = 0\) and local minima at \(x = -3\) and \(x = 2\).

Write down the x-intercepts of the graph of the derivative function, \(f^\prime\).
Write down all values of \(x\) for which \(f^\prime(x)\) is positive.
At point \(D\) on the graph of \(f\), the x-coordinate is \(-0.5\).

Explain why \(f^{\prime\prime}(x) < 0\) at \(D\).

IB Calculus Problem 11

2026-05-19T02:12:44+00:00

Consider the function \(f\) whose second derivative is \(f^{\prime\prime}(x) = 3x -1\).

The graph of \(f\) has a minimum point at \(A(2 ; 4)\) and a maximum point at \(B(-\frac{4}{3}; \frac{358}{27})\).

Use the second derivative to justify that \(B\) is a maximum.
Given that \(f^\prime(x) = \frac{3}{2}x^2 - x + p\), show that \(p = -4\).
Find \(f(x)\).

IB Calculus Problem 12

2026-05-19T02:07:34+00:00

Let \(f(x) = 6 + 6\,sin\,x\).

Part of the graph of \(f\) is shown below.

The figure is not to scale.

The shaded region is bounded by the curve of \(f\), the x-axis and the y-axis.

The path makes an angle of \(4^\circ\) with the horizontal.

Solve, for \(0 \leq x \leq 2\pi\)
1. \(6 + 6\,sin\,x = 6\).
2. \(6 + 6\,sin\,x = 0\).
Write down the exact value of the x-intercept of \(f\), for \(0 \leq x \leq 2\pi\).
The area of the shaded region is \(k\). Find the value of \(k\), giving your answer in terms of \(\pi\).

Let \(g(x) = 6 + 6\,sin\,(x - \frac{\pi}{2})\). The graph of \(f\) is transformed to that of \(g\).

Give a full geometric description of this transformation.
Given that \(\int_p^{p+\frac{3\pi}{2}}g(x)\,dx = k\) and \(0 \leq p < 2\pi\), write down the two values of \(p\).

IB Calculus Problem 13

2026-05-19T02:04:07+00:00

Let \(f^\prime(x) = 12x^2 - 2\).

Given that \(f(-1) = -1\), find \(f(x)\).

IB Calculus Problem 14

2026-05-19T02:00:55+00:00

The velocity \(v\), in \(ms^{-1}\), of a particle moving in a straight line is given by \(v=e^{3t-2}\), where \(t\) is the time in seconds.

Find the acceleration of the particle at time \(t = 1\).
For what value of \(t\) does the particle have a velocity of \(22.3ms^{-1}\)?
Find the distance travelled during the first second.

IB Calculus Problem 15

2026-05-18T20:11:31+00:00

Let \(f(x) = 3 Cos2x + Sin^2x\).

Show that \(f^\prime(x) = -5Sin2x\).
In the interval \(\frac{\pi}{4} \leq x \leq \frac{3\pi}{4}\), a normal to the graph of \(f\) has equation \(x = k\).

Find the value of \(k\).

IB Calculus Problem 16

2026-05-18T20:09:52+00:00

A particle \(P\) moves along a straight line. The velocity \(v\) \(ms^{-1}\) of \(P\) after \(t\) seconds is given by \(v(t) = 7 cos t - 5t^{cos t}\), for \(0 \leq t \leq 7\).

The following diagram shows the graph of \(v\).

The figure is not to scale.

Find the initial velocity of \(P\).
Find the maximum speed of \(P\).
Write down the number of times the acceleration of \(P\) is \(0\) \(ms^{-2}\).
Find the acceleration of \(P\) when the particle changes direction.
Find the total distance travelled by \(P\).

IB Calculus Problem 17

2026-05-18T20:07:39+00:00

Let \(f(x) = Cos(e^x)\), for \(-2 \leq x \leq 2\).

Find \(f^\prime(x)\).
On the set of axes below, sketch the graph of \(f^\prime(x)\).

IB Calculus Problem 18

2026-05-18T20:04:46+00:00

A particle moves along a straight line with velocity \(v = 12t - 2t^3 - 1\), for \(t \geq 0\), where \(v\) is in centimetres per second and \(t\) is in seconds.

Find the acceleration of the particle at time \(t = 2.7\) seconds.
Find the displacement of the particle after \(1.3\) seconds.

IB Calculus Problem 19

2026-05-18T20:02:49+00:00

Let \(f(x) = ax^3 + bx^2 + c\), where \(a\), \(b\) and \(c\) are real numbers. The curve of \(f\) passes through the point \(( 2; 9 )\).

Show that \(8a + 4b +c = 9\).

The curve of \(f\) has a local minimum at \((1; 4)\).

Find two more equations in terms of \(a\), \(b\) and \(c\), giving your answers in a form similar to that of part A.
Find the value of \(a\), the value of \(b\) and the value of \(c\).

IB Calculus Problem 20

2026-05-18T20:00:06+00:00

The gradient of a function is given by \(\dfrac{dy}{dx} = 10e^{2x - 5}\). When \(x = 0\), \(y = 8\).

Find the value of \(y\) when \(x = 1\).

IB Calculus Problem 21

2026-05-18T19:58:04+00:00

Let \(f(x) = e^x Sin 2x + 10\), with \(0 \leq x \leq 4\).

Part of the graph of \(f\) is shown below.

The figure is not to scale.

Shown are an x-intercept at point \(A\), a local maximum at point \(M\) with \(x = p\) and a local minimum at point \(N\) with \(x = q\).

State the x-coordinate of \(A\).
Find the value of
1. \(p\)
2. \(q\)
Find \(\int_p^qf(x)dx\).

Explain why this is not the area of the shaded region.

IB Calculus Problem 22

2026-05-18T19:55:55+00:00

Consider \(f(x) = x\ln(4 - x^2)\), with \(-2 < x < 2\).

Part of the graph of \(f\) is shown below.

Let \(P\) and \(Q\) be the points on the curve of \(f\) where the tangent to the graph of \(f\) is parallel to the x-axis.

1. Find the x-coordinates of \(P\) and \(Q\).
2. Consider \(f(x) = k\).
  
  State all values of \(k\) for which there are exactly two solutions.

Let \(g(x) = x^3\ln(4 - x^2)\), with \(-2 < x < 2\).

Show that \(g^\prime(x)=\frac{-2x^4}{4 - x^2} + 3x^2\ln(4 - x^2)\).
Sketch the graph of \(g^\prime\).
Consider \(g^\prime(x) = w\).

State all values of \(w\) for which there are exactly two solutions.