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Question 1

SLPaper 1

Consider the functions $a(x) = 11.5x^2$ and $b(x) = 13.2x^2 + 12.6$ .

1.

Determine the area between the curves $a(x)$ and $b(x)$ from $x = -1$ to $x = 1$ .

[2]

Question 2

HLPaper 1

Consider the function $f(x) = x^3 - 3x^2 + 4$ on the interval $[0, 2]$ .

1.

Calculate the volume of the solid formed by revolving the region under the curve $f(x)$ from $x = 0$ to $x = 2$ about the $x$ -axis.

[3]

2.

Determine the volume of the solid formed by revolving the region under the curve $f(x)$ from $x = 0$ to $x = 2$ about the $y$ -axis.

[3]

Question 3

HLPaper 1

A particle moves along a straight line with its position at time $t$ given by the function $s(t) = t^3 - 6t^2 + 9t + 2$ , where $s$ is in meters and $t$ is in seconds.

1.

Find the velocity of the particle at time $t$ .

[2]

2.

Determine the acceleration of the particle at time $t$ .

[2]

3.

Find the time at which the particle is momentarily at rest.

[3]

4.

Calculate the total distance traveled by the particle from $t = 0$ to $t = 4$ .

[3]

Question 4

HLPaper 1

Consider the function $f(x) = 3x^2 - 2x + 1$ .

1.

Find the indefinite integral of $f(x)$ with respect to $x$ .

[5]

2.

Evaluate the definite integral of $f(x)$ from $x = 1$ to $x = 3$ .

[6]

Question 5

HLPaper 1

Consider the function $f(x) = x e^x$ .

1.

Use integration by parts to find the integral of $f(x) = x e^x$ with respect to $x$ .

[2]

Question 6

HLPaper 1

Consider the function $h(x) = x^2 \sin(x)$ .

1.

Use integration by parts to find the integral of $h(x) = x^2 \sin(x)$ with respect to $x$ .

[4]

Question 7

HLPaper 2

Consider the function $f(x) = 121.5x^2 - 486x + 364.5$ .

1.

Find the area between the curve $y = f(x)$ and the x-axis from $x = 1$ to $x = 3$ . Use your GDC to evaluate the integral.

[2]

Question 8

HLPaper 1

Consider the integral $\int (3x^2 + 2x) e^{x^2 + x} \, dx$ . Use the substitution $u = x^2 + x$ .

1.

Find the integral $\int (3x^2 + 2x) e^{x^2 + x} \, dx$ using the substitution $u = x^2 + x$ .

[3]

Question 9

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ ,.

1.

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

2.

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

3.

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 10

HLPaper 1

A particle moves along a straight line with its acceleration given as a function of time.

1.

Given the acceleration of the particle is $a(t) = 3t^2 - 2t + 1$ , find the velocity function $v(t)$ if the initial velocity $v(0) = 4$ .

[2]

2.

Using the velocity function $v(t) = t^3 - t^2 + t + 4$ , find the displacement function $s(t)$ if the initial displacement $s(0) = 0$ .

[2]

Calculus

Calculus