Calculus Volume 3

# Review Exercises

Calculus Volume 3Review Exercises

### Review Exercises

True or False? Justify your answer with a proof or a counterexample.

202 .

A parametric equation that passes through points P and Q can be given by $r(t)=〈t2,3t+1,t−2〉,r(t)=〈t2,3t+1,t−2〉,$ where $P(1,4,−1)P(1,4,−1)$ and $Q(16,11,2).Q(16,11,2).$

203 .

$d d t [ u ( t ) × u ( t ) ] = 2 u ′ ( t ) × u ( t ) d d t [ u ( t ) × u ( t ) ] = 2 u ′ ( t ) × u ( t )$

204 .

The curvature of a circle of radius $rr$ is constant everywhere. Furthermore, the curvature is equal to $1/r.1/r.$

205 .

The speed of a particle with a position function $r(t)r(t)$ is $(r′(t))/(|r′(t)|).(r′(t))/(|r′(t)|).$

Find the domains of the vector-valued functions.

206 .

$r ( t ) = 〈 sin ( t ) , ln ( t ) , t 〉 r ( t ) = 〈 sin ( t ) , ln ( t ) , t 〉$

207 .

$r ( t ) = 〈 e t , 1 4 − t , sec ( t ) 〉 r ( t ) = 〈 e t , 1 4 − t , sec ( t ) 〉$

Sketch the curves for the following vector equations. Use a calculator if needed.

208 .

[T] $r(t)=〈t2,t3〉r(t)=〈t2,t3〉$

209 .

[T] $r(t)=〈sin(20t)e−t,cos(20t)e−t,e−t〉r(t)=〈sin(20t)e−t,cos(20t)e−t,e−t〉$

Find a vector function that describes the following curves.

210 .

Intersection of the cylinder $x2+y2=4x2+y2=4$ with the plane $x+z=6x+z=6$

211 .

Intersection of the cone $z=x2+y2z=x2+y2$ and plane $z=y−4z=y−4$

Find the derivatives of $u(t),u(t),$ $u′(t),u′(t),$ $u′(t)×u(t),u′(t)×u(t),$ $u(t)×u′(t),u(t)×u′(t),$ and $u(t)·u′(t).u(t)·u′(t).$ Find the unit tangent vector.

212 .

$u ( t ) = 〈 e t , e − t 〉 u ( t ) = 〈 e t , e − t 〉$

213 .

$u ( t ) = 〈 t 2 , 2 t + 6 , 4 t 5 − 12 〉 u ( t ) = 〈 t 2 , 2 t + 6 , 4 t 5 − 12 〉$

Evaluate the following integrals.

214 .

$∫ ( tan ( t ) sec ( t ) i − t e 3 t j ) d t ∫ ( tan ( t ) sec ( t ) i − t e 3 t j ) d t$

215 .

$∫14u(t)dt,∫14u(t)dt,$ with $u(t)=〈ln(t)t,1t,sin(tπ4)〉u(t)=〈ln(t)t,1t,sin(tπ4)〉$

Find the length for the following curves.

216 .

$r(t)=〈3t,4cos(t),4sin(t)〉r(t)=〈3t,4cos(t),4sin(t)〉$ for $1≤t≤41≤t≤4$

217 .

$r(t)=2i+tj+3t2kr(t)=2i+tj+3t2k$ for $0≤t≤10≤t≤1$

Reparameterize the following functions with respect to their arc length measured from $t=0t=0$ in direction of increasing $t.t.$

218 .

$r ( t ) = 2 t i + ( 4 t − 5 ) j + ( 1 − 3 t ) k r ( t ) = 2 t i + ( 4 t − 5 ) j + ( 1 − 3 t ) k$

219 .

$r ( t ) = cos ( 2 t ) i + 8 t j − sin ( 2 t ) k r ( t ) = cos ( 2 t ) i + 8 t j − sin ( 2 t ) k$

Find the curvature for the following vector functions.

220 .

$r ( t ) = ( 2 sin t ) i − 4 t j + ( 2 cos t ) k r ( t ) = ( 2 sin t ) i − 4 t j + ( 2 cos t ) k$

221 .

$r ( t ) = 2 e t i + 2 e − t j + 2 t k r ( t ) = 2 e t i + 2 e − t j + 2 t k$

222 .

Find the unit tangent vector, the unit normal vector, and the binormal vector for $r(t)=2costi+3tj+2sintk.r(t)=2costi+3tj+2sintk.$

223 .

Find the tangential and normal acceleration components with the position vector $r(t)=〈cost,sint,et〉.r(t)=〈cost,sint,et〉.$

224 .

A Ferris wheel car is moving at a constant speed $vv$ and has a constant radius $r.r.$ Find the tangential and normal acceleration of the Ferris wheel car.

225 .

The position of a particle is given by $r(t)=〈t2,ln(t),sin(πt)〉,r(t)=〈t2,ln(t),sin(πt)〉,$ where $tt$ is measured in seconds and $rr$ is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

The following problems consider launching a cannonball out of a cannon. The cannonball is shot out of the cannon with an angle $θθ$ and initial velocity $v0.v0.$ The only force acting on the cannonball is gravity, so we begin with a constant acceleration $a(t)=−gj.a(t)=−gj.$

226 .

Find the velocity vector function $v(t).v(t).$

227 .

Find the position vector $r(t)r(t)$ and the parametric representation for the position.

228 .

At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel?

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