Sec. 2.2 - Sine and Cosine Functions

Overview

This section covers the following concepts: Better understanding of the derivative of an exponential function, derivatives of the sine and cosine functions.

Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

• Know the graph of a general exponential function \(f(x) = a^x\) and its most important features (y-intercept, behaviour for large positive or negative x-values) - see http://mathquest.carroll.edu/CarrollActiveCalculus/S_0_2_Exponentials.html for review.

• Be able to use the limit definition of the derivative to see why it’s hard to directly compute the derivative of an exponential function \(f(x) = a^x\), where \(a > 1\)

• Understand why it is reasonable that for \(f(x) = a^x\), its derivative should be of the form \(f'(x) = a^x \cdot c\), for some positive constant \(c\).

• Know the general rule for \(\frac{d}{dx}[a^x]\).

• Understand why the function \(e^x\) is very special when it comes to its relationship to its own derivative.

• State the values of \(\sin x\) and \(\cos x\) at the angle values \(0\), \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\) and other related points on the unit circle without a calculator. (See http://mathquest.carroll.edu/CarrollActiveCalculus/S_0_5_TrigFunctions.html for review.)

Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

• Know the derivatives of the functions \(y = \sin(x)\) and \(y = \cos(x)\).

• Use the derivatives of \(y = \sin(x)\) and \(y = \cos(x)\) in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.

• Prove that \(\frac{d}{dx}\left[ \sin x \right] = \cos x\) using the fact that \(\displaystyle \lim_{h\to 0} \frac{\sin x}{x} = 1\) and that \(\sin (x+h) = \sin x\cos h + \cos x \sin h\), which need not be proven. The proof appears on this appendix page. Make sure to understand where in the proof we need \(x\) to be measured in radians for the formula to be valid.

To prepare for class

• Read all of section 0.2 at http://mathquest.carroll.edu/CarrollActiveCalculus/S_0_2_Exponentials.html, to review pre-calculus facts about exponential functions.

• Use Desmos to do the following:

  • Enter the function \(f(x)=a^x\), and create a slider for the parameter “\(a\)” (when it asks you if you want that).
  • Click on one of the two arrows to the left or right of the slider, and use the dialogue box to set the slider to run from 1.1 to 5.5 with increment 0.0001.
  • Explore the overall situation by moving the slider to change the value of \(a\), then write down (on paper) at least three different things you observe - and include these comments in your Feedback for this class.

• Read the introduction of section 2.2 in Active Calculus, up to the preview activity.

• Do the Preview Activity for section 2.2 (on WeBWorK if required by your teacher)

• Read all of section 0.5 at http://mathquest.carroll.edu/CarrollActiveCalculus/S_0_5_TrigFunctions.html, to review pre-calculus facts about the trigonometric functions \(\sin(x)\), \(\cos(x)\), \(\tan(x)\) and their inverse functions, as well as about measuring angles in radians.

• Use Desmos to experiment with the family of functions \(f(x)=A\sin(Bx)\), using sliders for the parameters \(A\) and \(B\). Play around with different values for \(A\) and \(B\), and see how this affects the graph. Then repeat the same for \(g(x)=C\cos(Dx)\) (in the same picture as \(f(x)\)). Write down (on paper) at least three different things you observe - and include these comments in your Feedback for this class.

• Watch these videos to review the values for special angles for sine and cosine and how to visualize them on the unit circle:

After class

• Watch the first part of the following video (until 3:15):

• Do some experimentation with the following interactive applets:

  • An Important Trig Limit \(\displaystyle \lim_{\theta\to 0} \frac{\sin(\theta)}{\theta} = 1\)

  • The Derivative of Elementary Functions

• Prove (using the limit definition of the derivative) that \(\frac{d}{dx}\left[ \sin x \right] = \cos x\). Use the fact that \(\displaystyle \lim_{h\to 0} \frac{\sin x}{x} = 1\) and that \(\sin (x+h) = \sin x\cos h + \cos x \sin h\), which need not be proven. After attempting it yourself, read (and try to understand all steps of) the proof shown on this appendix page. Make sure to understand where in the proof we need \(x\) to be measured in radians for the formula to be valid.

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