## MATH1111 Quizzes

Differentiating Trigonometric Quiz
Web resources available Questions

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 Learning Hub (Mathematics) 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/$\sim$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.

Which of the following statements are correct? (Zero or more options can be correct)
 a) If $f\left(x\right)=sin2x$ then ${f}^{\prime }\left(x\right)=2cos2x\phantom{\rule{0.3em}{0ex}}.$ b) If $f\left(x\right)=sin3x$ then ${f}^{\prime }\left(x\right)=-3cos3x\phantom{\rule{0.3em}{0ex}}.$ c) If $f\left(x\right)=2cosx$ then ${f}^{\prime }\left(x\right)=sin2x\phantom{\rule{0.3em}{0ex}}.$ d) If $f\left(x\right)=3cosx$ then ${f}^{\prime }\left(x\right)=-3sinx\phantom{\rule{0.3em}{0ex}}.$ e) If $f\left(x\right)=2tanx$ then ${f}^{\prime }\left(x\right)={sec}^{2}2x\phantom{\rule{0.3em}{0ex}}.$

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 $f\left(x\right)=sin3x$ then ${f}^{\prime }\left(x\right)=3cos3x\phantom{\rule{0.3em}{0ex}}.$
There is at least one mistake.
For example, choice (c) should be False.
If $f\left(x\right)=2cosx$ then ${f}^{\prime }\left(x\right)=-2sinx\phantom{\rule{0.3em}{0ex}}.$
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 $f\left(x\right)=2tanx$ then ${f}^{\prime }\left(x\right)=2{sec}^{2}x\phantom{\rule{0.3em}{0ex}}.$
Correct!
1. True
2. False If $f\left(x\right)=sin3x$ then ${f}^{\prime }\left(x\right)=3cos3x\phantom{\rule{0.3em}{0ex}}.$
3. False If $f\left(x\right)=2cosx$ then ${f}^{\prime }\left(x\right)=-2sinx\phantom{\rule{0.3em}{0ex}}.$
4. True
5. False If $f\left(x\right)=2tanx$ then ${f}^{\prime }\left(x\right)=2{sec}^{2}x\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the derivative of  $f\left(x\right)=cos\left({x}^{2}+3x\right)\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) ${f}^{\prime }\left(x\right)=sin\left(2x+3\right)$ b) ${f}^{\prime }\left(x\right)=-sin\left(2x+3\right)$ c) ${f}^{\prime }\left(x\right)=-\left(2x+3\right)sin\left({x}^{2}+3x\right)$ d) ${f}^{\prime }\left(x\right)=\left(2x+3\right)sin\left({x}^{2}+3x\right)$

Choice (a) is incorrect
Try again, you have not applied the chain rule correctly.
Choice (b) is incorrect
Try again, you have not applied the chain rule correctly.
Choice (c) is correct!
${f}^{\prime }\left(x\right)=\left(2x+3\right)×-sin\left({x}^{2}+3x\right)=-\left(2x+3\right)sin\left({x}^{2}+3x\right)$
Choice (d) is incorrect
Try again, watch the sign.
Which of the following is the derivative of  $x=2sin3t+{cos}^{2}t\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) $\frac{dx}{dt}=6cos3t-2sintcost$ b) $\frac{dx}{dt}=6cos3t-2cost$ c) $\frac{dx}{dt}=6cos3t{cos}^{2}t-2sintcostsin3t$ d) $\frac{dx}{dt}=6cos3t{cos}^{2}t-2costsin3t$

Choice (a) is correct!
$\frac{dx}{dt}=2×3cost+2cost×-sint=6cos3t-2sintcost$
Choice (b) is incorrect
Try again, you have not used the chain rule correctly for the second term.
Choice (c) is incorrect
Try again, you have successfully found the derivative of  $x=2sin3t×{cos}^{2}t\phantom{\rule{0.3em}{0ex}}.$
Choice (d) is incorrect
Try again, you have seem to be trying to find the derivative of  $x=2sin3t×{cos}^{2}t\phantom{\rule{0.3em}{0ex}}.$
Which of the following is the derivative of  $h\left(s\right)=tan3s+\frac{{s}^{3}+3sins}{cos4s}\phantom{\rule{0.3em}{0ex}}?$ Exactly one option must be correct)
 a) ${h}^{\prime }\left(s\right)={sec}^{2}3s+\frac{\left(3{s}^{2}+3coss\right)cos4s-\left({s}^{3}+3sins\right)4sin4s}{{cos}^{2}4s}$ b) ${h}^{\prime }\left(s\right)={sec}^{2}3s+\frac{\left(3{s}^{2}+3coss\right)cos4s+\left({s}^{3}+3sins\right)4sin4s}{{cos}^{2}4s}$ c) ${h}^{\prime }\left(s\right)=3{sec}^{2}3s+\frac{\left(3{s}^{2}+3coss\right)cos4s-\left({s}^{3}+3sins\right)4sin4s}{{cos}^{2}4s}$ d) None of the above.

Choice (a) is incorrect
Try again, you have not differentiated the first term correctly, you may have a sign problem as well.
Choice (b) is incorrect
Try again, you have not differentiated the first term correctly.
Choice (c) is incorrect
Try again, you have not differentiated the second term correctly.
Choice (d) is correct!
${h}^{\prime }\left(s\right)=3{sec}^{2}3s+\frac{\left(3{s}^{2}+3coss\right)cos4s+\left({s}^{3}+3sins\right)4sin4s}{{cos}^{2}4s}$