r(t) = ti + 3tj + t²k, u(t) = 4ti + t²j + t³k Use the properties of the derivative to find the following. (a)r′(t) (b)d/dt [3r(t) - u(t)] (c)d/dt (5t)u(t) (d)d/dt [r(t)·u(t)] (e)d/dt [r(t)×u(t)] (f)d/dt r(2t)

Match the equation. r(t) = ti + 2tj + t²k, -2 ≤ t ≤ 2

Evaluate the definite integral.

$$∫_0^3 ǁti + t²jǁ dt$$

Simplify each expression.

$$( 2 x ) ^ { 3 } x ^ { 2 }$$

Evaluate the definite integral. ∫_0^π/4 [(sec t tan t)i + (tan t)j + (2 sin t cos t)k] dt

A pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as t increases. (b) Find a rectangular-coordinate equation for the curve by eliminating the parameter. x = 2t, y = t + 6

Use a calculator to evaluate each expression. Round to the nearest ten-thousandth.

$$\log 12$$

Find r′(t). r(t) = ⟨arcsin t, arccos t, 0⟩

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Different parametrizations of the same curve result in identical tangent vectors at a given point on the curve.

Find the domain of the vector-valued function. r(t) = F(t) x G(t), where F(t) = t³i - tj + tk, G(t) = ∛t i + 1 / t+1 j + (t+2)k

The 2014 Ford Mustang gets about 13.18 km/l on the highway. The 2014 Ford Mustang GT gets about 10.63 km/l. (a) What is the highway gas mileage of the Ford Mustang in meters per liter? (b) What is the highway gas mileage of the Ford Mustang GT in dekameters per liter? (c) How much better mileage, in kilometers per liter, does the Mustang get than the Mustang GT?

Find r′(t). r(t) = 4√t i + t² √t j + ln t²k

Find the domain of the vector-valued function. r(t) = F(t) x G(t), where F(t) = sin ti + cos tj, G(t) = sin tj + cos tk

Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = t²i + t³j