3.5E: Trig Derivatives Exercises

3.5: Derivatives of Trigonometric Functions

Exercise:

For the following exercises, find $\frac{dy}{dx}$ for the given functions.

175) $y=x^2−secx+1$

Answer:

$\frac{dy}{dx}=2x−secxtanx$

176) $y=3cscx+\frac{5}{x}$

177) $y=x^2cotx$

Answer:

$\frac{dy}{dx}=2xcotx−x^2csc^2x$

178) $y=x−x^3sinx$

179) $y=\frac{secx}{x}$

Answer:

$\frac{dy}{dx}=\frac{xsecxtanx−secx}{x^2}$

180) $y=sinxtanx$

181) $y=(x+cosx)(1−sinx)$

Answer:

$\frac{dy}{dx}=(1−sinx)(1−sinx)−cosx(x+cosx)$

182) $y=\frac{tanx}{1−secx}$

183) $y=\frac{1−cotx}{1+cotx}$

Answer:

$\frac{dy}{dx}=\frac{2csc^2x}{(1+cotx)^2}$

184) $y=cosx(1+cscx)$

For the following exercises, find the equation of the tangent line to each of the given functions at the indicated values of $x$. Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

185) $[T] f(x)=−\sin{x},x=0$

Answer:

$y=−x$

186) $[T] f(x)=cscx,x=\frac{π}{2}$

187) $[T] f(x)=1+cosx,x=\frac{3π}{2}$

Answer:

$y=x+\frac{2−3π}{2}$

188) $[T] f(x)=secx,x=\frac{π}{4}$

189) $[T] f(x)=x^2−\tan{x}=0$

Answer:

$y=−x$

190) $[T] f(x)=5cotxx=\frac{π}{4}$

For the following exercises, find $\frac{d^2y}{dx^2}$ for the given functions.

191) $y=xsinx−cosx$

Answer:

$3cosx−xsinx$

192) $y=sinxcosx$

193) $y=x−\frac{1}{2}sinx$

Answer:

$\frac{1}{2}sinx$

194) $y=\frac{1}{x}+tanx$

195) $y=2cscx$

Answer:

$csc(x)(3csc^2(x)−1+cot^2(x))$

196) $y=sec^2x$

197) Find all $x$ values on the graph of $f(x)=−3sinxcosx$ where the tangent line is horizontal.

Answer:

$\frac{(2n+1)π}{4}$,where $n$ is an integer

198) Find all $x$ values on the graph of $f(x)=x−2cosx$ for $0<x<2π$ where the tangent line has slope 2.

199) Let $f(x)=cotx.$ Determine the points on the graph of $f$ for $0<x<2π$ where the tangent line(s) is (are) parallel to the line $y=−2x$.

Answer:

$(\frac{π}{4},1),(\frac{3π}{4},−1)$

200) [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $s(t)=−6cost$ where s is measured in inches and t is measured in seconds. Find the rate at which the spring is oscillating at $t=5$ s.

201) Let the position of a swinging pendulum in simple harmonic motion be given by $s(t)=acost+bsint$. Find the constants $a$ and $b$ such that when the velocity is 3 cm/s, $s=0$ and $t=0$.

Answer:

$a=0,b=3$

202) After a diver jumps off a diving board, the edge of the board oscillates with position given by $s(t)=−5cost$ cm at $t$seconds after the jump.

a. Sketch one period of the position function for $t≥0$.

b. Find the velocity function.

c. Sketch one period of the velocity function for $t≥0$.

d. Determine the times when the velocity is 0 over one period.

e. Find the acceleration function.

f. Sketch one period of the acceleration function for $t≥0$.

203) The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5sinx$ where $y$ is the number of hamburgers sold and x represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $y'$ and determine the intervals where the number of burgers being sold is increasing.

Answer:

$y′=5cos(x)$, increasing on $(0,\frac{π}{2}),(\frac{3π}{2},\frac{5π}{2})$, and $(\frac{7π}{2},12)$

204) [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $y(t)=0.5+0.3cost$, where t is months since January. Find $y′$and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises, use the quotient rule to derive the given equations.

205) $\frac{d}{dx}(cotx)=−csc^2x$

206) $\frac{d}{dx}(secx)=secxtanx$

207) $\frac{d}{dx}(cscx)=−cscxcotx$

208) Use the definition of derivative and the identity $cos(x+h)=cosxcosh−sinxsinh$

to prove that $\frac{d(cosx)}{dx}=−sinx$.

For the following exercises, find the requested higher-order derivative for the given functions.

209) $\frac{d^3y}{dx^3}$ of $y=3cosx$

Answer:

$3sinx$

210) $\frac{d^2y}{dx^2}$ of $y=3sinx+x^2cosx$

211) $\frac{d^4y}{dx^4}$ of $y=5cosx$

Answer:

$5cosx$

212) $\frac{d^2y}{dx^2}$ of $y=secx+cotx$

213) $\frac{d^3y}{dx^3}$ of $y=x^{10}−secx$

Answer:

$720x^7−5tan(x)sec^3(x)−tan^3(x)sec(x)$

ADD IN J214 - J217