Component

Component: Geometrical applications of differentiation

Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3

Question 2

Question 2

Written Paper Section I Question 2 - 2002 HSC

Find the equation of a tangent to a curve. Differentiate and integrate functions. Use the sine rule to find the exact value of a ratio.

Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3

Question 5

Question 5

Written Paper Section I Question 5 - 2010 HSC

Show that the surface area of a cylinder has a minimum value for a particular value of its radius. Prove trigonometric identities and find the exact value of an integral. Find the ordinates at two x-values for areas under a curve enclosed by the x-axis, a given ordinate and the ordinates to be found, and the curve.

Student Responses (13)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2

Question 6

Question 6

Written Paper Section I Question 6 - 2010 HSC

Show that a specified graph has no stationary points. Determine values for which the graph is concave down and for which it is concave up. Sketch the graph. Find an angle in radians, the length of an interval, and the area of a shaded region. Prove two specified triangles are congruent.

Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3

Question 6

Question 6

Written Paper Section I Question 6 - 2002 HSC

Sketch a function representing a semi-circle and state the range of the function. Given the gradient function of a curve, determine the equation of the curve. Sketch the curve, labelling turning points and the y-intercept. Determine for what values of the independent variable x the curve is concave up. Calculate the volume of a container formed by rotating part of a given curve.

Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3

Question 6

Question 6

Written Paper Section I Question 6 - 2001 HSC

Calculate a term and sum of the given arithmetic series. Find the decimal value of a pronumeral in an alternative expression of an exponential equation. Find the coordinates of two stationary points on a given curve. Determine the values of x for which the curve is concave up and give reasons for the answer. Find the possible values of a constant for which an equation related to the curve has three real solutions.

Student Responses (11)

Band 5/6 response 1 Band 5/6 response 3 Band 5/6 response 2 Band 4/5 response 1 Band 4/5 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 2/3 response 1 Band 2/3 response 2 Band 1/2 response 1 Band 1/2 response 2

Question 8

Question 8

Written Paper Section I Question 8 - 2010 HSC

Calculate a population that is growing exponentially. Determine the probability that two coins tossed together will both show tails. Find values that determine the equation of a sine graph. Draw a graph on the same set of axes as the given graph. Find the values of a coefficient in the equation of a function for which the function is an increasing function.

Student Responses (15)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1 Band 1/2 response 2 Band 1/2 response 3

Question 8

Question 8

Written Paper Section I Question 8 - 2001 HSC

Calculate values of constants in equation modelling exponential population growth. Find the probability of a single-stage event and of a multi-stage event. Determine the maximum and minimum values of the rate of change of a quantity. Sketch the graph of the quantity as a function of time and identify any points on the graph where the concavity changes.

Student Responses (14)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 2 Band 3/4 response 1 Band 3/4 response 2 Band 2/3 response 2 Band 2/3 response 1 Band 2/3 response 2 Band 1/2 response 1 Band 1/2 response 2

Question 9

Question 9

Written Paper Section I Question 9 - 2010 HSC

Apply geometric series to financial situations. Determine the values for which a given function is increasing and at its maximum. Find a further value of the function. Draw a graph of the function.

Student Responses (13)

Band 5/6 response 1 Band 5/6 response 2 Band 5/6 response 3 Band 4/5 response 1 Band 4/5 response 2 Band 4/5 response 3 Band 3/4 response 1 Band 3/4 response 2 Band 3/4 response 3 Band 2/3 response 1 Band 2/3 response 2 Band 2/3 response 3 Band 1/2 response 1

Question 10

Question 10

Written Paper Section I Question 10 - 2002 HSC

Verify equations related to sectors of a circle and graph a related piecemeal function. Differentiate a complex expression. Verify changes in a quantity and describe a related situation, giving reasons for the answer.

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