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MathQuiz 4.3
University of Sydney
School of Mathematics
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© Copyright 2004-2006

Differentiating Trigonometric Quiz

Question

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This quiz tests the work covered in Lecture 16 and corresponds to Section 3.5 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There are more web quizzes at Wiley, select Section 5.

Section 3.7 of The Mathematics Learning Centre’s booklet on differentiation Introduction to Differential Calculus covers differentiating trigonometric functions.

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/trig2.html gives another explanation of the derivative of the basic trigonometric functions.

There is an applet at http://www.ies.co.jp/math/java/calc/sin_diff/sin_diff.html which graphs the derivatives by looking at the tangent to the curve.

You might want to go back to http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/chainruledirectory/ChainRule.html or http://archives.math.utk.edu/visual.calculus/2/chain_ rule.2/ and try the trigonometrical problems if you haven’t already.

Question 1

Which of the following statements are correct?

There is at least one mistake.
For example, choice (a) should be true.

There is at least one mistake.
For example, choice (b) should be false.

If $f(x) = sin3x$ then $f′(x) = 3cos3x.$

There is at least one mistake.
For example, choice (c) should be false.

If $f(x) = 2cosx$ then $f′(x) = - 2 sinx .$

There is at least one mistake.
For example, choice (d) should be true.

There is at least one mistake.
For example, choice (e) should be false.

If $f(x) = 2 tan x$ then $f′(x) = 2 sec2x .$

Your answers are correct

True.
False. If $f(x) = sin3x$ then $f′(x) = 3cos3x.$
False. If $f(x) = 2cosx$ then $f′(x) = - 2 sinx .$
True.
False. If $f(x) = 2 tan x$ then $f′(x) = 2 sec2x .$

Question 2

Which of the following is the derivative of $f (x) = cos(x2 + 3x)?$

Not correct. Choice (a) is false.

Try again, you have not applied the chain rule correctly.

Not correct. Choice (b) is false.

Try again, you have not applied the chain rule correctly.

Your answer is correct.

$f′(x) = (2x+ 3)× - sin(x2 + 3x) = - (2x+ 3)sin(x2 + 3x)$

Not correct. Choice (d) is false.

Try again, watch the sign.

Question 3

Which of the following is the derivative of $x = 2 sin3t+ cos2t?$

Your answer is correct.

$dx= 2 × 3cost+ 2cost× - sin t = 6cos3t- 2sin tcost dt$

Not correct. Choice (b) is false.

Try again, you have not used the chain rule correctly for the second term.

Not correct. Choice (c) is false.

Try again, you have successfully found the derivative of $2 x = 2sin 3t× cos t.$

Not correct. Choice (d) is false.

Try again, you have seem to be trying to find the derivative of $x = 2 sin3t× cos2t.$

Question 4

Which of the following is the derivative of $h(s) = tan3s+ s3 +-3sin-s? cos4s$

Not correct. Choice (a) is false.

Try again, you have not differentiated the first term correctly, you may have a sign problem as well.

Not correct. Choice (b) is false.

Try again, you have not differentiated the first term correctly.

Not correct. Choice (c) is false.

Try again, you have not differentiated the second term correctly.

Your answer is correct.

$′ 2 (3s2 +-3coss)cos4s+-(s3 +-3sin-s)4-sin4s h (s) = 3sec 3s+ cos2 4s$