- State the value of
. (1 mark) --- 1 WORK AREA LINES (style=lined) ---
- The derivative,
, can be expressed in the form . - Find the real number
. (1 mark) --- 2 WORK AREA LINES (style=lined) ---
-
i. Let
be a real number. Find, in terms of , the equation of the tangent to at the point . (1 mark) ii. Hence, or otherwise, find the equation of the tangent to--- 3 WORK AREA LINES (style=lined) ---
that passes through the origin, correct to three decimal places. (2 marks) --- 8 WORK AREA LINES (style=lined) ---
Let
- Find the coordinates of the point of inflection for
, correct to two decimal places. (1 mark) --- 2 WORK AREA LINES (style=lined) ---
- Find the largest interval of
values for which is strictly decreasing. - Give your answer correct to two decimal places. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Apply Newton's method, with an initial estimate of
, to find an approximate -intercept of . - Write the estimates
and in the table below, correct to three decimal places. (2 marks)
--- 0 WORK AREA LINES (style=lined) ---
- For the function
, explain why a solution to the equation should not be used as an initial estimate in Newton's method. (1 mark) --- 3 WORK AREA LINES (style=lined) ---
- There is a positive real number
for which the function has a local minimum on the -axis. - Find this value of
. (2 marks) --- 5 WORK AREA LINES (style=lined) ---
Show Worked Solution
a.
| b. | ||
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ci.
♦♦ Mean mark (c)(i) 50%.
cii.
♦♦♦ Mean mark (c)(ii) 30%.
MARKER’S COMMENT: Some students did not substitute (0, 0) into the correct equation or did not find the value of
MARKER’S COMMENT: Some students did not substitute (0, 0) into the correct equation or did not find the value of
| d. | ||
e.
♦♦ Mean mark (e) 40%.
MARKER’S COMMENT: Round brackets were often used which were incorrect as endpoints were included. Some responses showed the interval where the function was strictly increasing.
MARKER’S COMMENT: Round brackets were often used which were incorrect as endpoints were included. Some responses showed the interval where the function was strictly increasing.
f.
g.
♦♦♦ Mean mark (g) 20%.
h.
♦♦♦ Mean mark (h) 10%.
MARKER’S COMMENT: Many students showed that
MARKER’S COMMENT: Many students showed that