SECTION 3.3 DERIVATIVES OF TRIG FUNCTIONS

1. Fill in the table below.

Derivatives of Trigonometric Functions:

• ddx(sinx) =

• ddx(cosx) =

• ddx(tanx) =

• ddx(cscx) =

• ddx(secx) =

• ddx(cotx) =

2. Find the derivative of y =secx

1� x tanx.

3. If f(✓) = e✓ sin(✓), find f 00(✓). Simplify your answers.

4. Findd

dt[t sin t cos t].

UAF Calculus I 1 3-3 Derivatives of Trig Functions

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5. An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the massis pulled down 2 cm past its rest position and then released, it vibrates vertically. The equation ofmotion is

s = 2 cos t+ 3 sin t, for t � 0,

where s is measured in centimeters and t is measured in seconds. (We are taking the positivedirection to be downward.)

(a) Why might you expect to use sines and cosines to model this particular problem?

(b) Find s(0), s0(0), and s00(0) including units.

(c) What do the numbers from part (a) indicate about the mass in the context of the problem?

6. A 12 foot ladder rests against a wall. Let ✓ be the angle between the ladder and the wall and let xbe the distance from the base of the ladder and the wall.

(a) Compute x as a function of ✓. (Drawing a picture will help.)

(b) How fast does x change with respect to ✓ when ✓ = ⇡/6? (Get an exact answer and a decimalapproximation.)

(c) Interpret your answer from part (b) in the context of the problem. (Units will help you here.)

(d) Determine how far the ladder is from the wall when ✓ = ⇡/6.

UAF Calculus I 2 3-3 Derivatives of Trig Functions

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Thedisplacement at time t o is 2cmThe initial velocity is 3curlsThe initial acceleration is 2cms2

TY Fp sin0 x 12 since µ12cos0 so d o

12 Ws 1275 6538 10.39

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When 0 476 Xe 12 Sir 6 12 12 6The ladder is 6ft from the wall Check

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