Sketch the space curve and find its length over the given interval. Vector-Valued Function: r(t) = ⟨2 sin t, 5t, 2 cos t⟩

(a) find the point on the curve at which the curvature K is a maximum, and (b) find the limit of K as x→∞. y = ln x

Find the curvature K of the curve, where s is the arc length parameter. r(s) = (1 + √2/2 s)i + (1 - √2/2 s)j

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

If x varies directly as y, find x when y = 8. x = 6 when y = 32

Determine which of the reflexive, symmetric, and transitive properties are satisfied by the given relation $R$ defined on set $S$.

$S$ is the set of all graduates of Michigan State University and $x R y$ means that $x$ first attended Michigan State University in the same year as $y$.

Find the length of the curve. $r(t)=\lang{t,\ 3cost, 3sint \rang},\ -5\leq t \leq 5$

Find $\mathbf{r}(t) \cdot \mathbf{u}(t)$. Is the result a vector-valued function? Explain.

$$\mathbf{r}(t)=(3 t-1) \mathbf{i}+\frac{1}{4} t^3 \mathbf{j}+4 \mathbf{k}$$

$$\mathbf{u}(t)=t^2 \mathbf{i}-8 \mathbf{j}+t^3 \mathbf{k}$$

Verify that the function $M(t)=K\left(\frac{M_{0}}{K}\right)^{\exp (-r t)}$ satisfies the properties $M(0)=M_{0}$ and $\lim _{t \rightarrow \infty} M(t)=K$.

Identify each of the following as either an expression or an equation.

$$9 r+3(r-4)=2$$

If x varies directly as y, find x when y = 8.

$$x=-4$$

when

$$y = 10$$

Graph the ordered pairs in a coordinate plane. Do you think that graph shows that the quantities vary directly? Explain your reasoning.

$$(-1,-1),(0,0),(1,1),(2,2)$$

Find the length and sketch the curve. r(t)=[4 cos t, 4 sin t, 5t] from (4, 0, 0) to (4, 0, 10π)

Write a true sentence using either $<$ or $>$. \

$$-\frac{14}{17}\ \square\ -\frac{27}{35}$$