91Ó°ÊÓ

A particle moves along the \(x\) -axis at a velocity of \(v(t)=1 / \sqrt{t}\) , \(t>0 .\) At time \(t=1,\) its position is \(x=4 .\) Find the acceleration and position functions for the particle.

A particle, initially at rest, moves along the \(x\) -axis such thatits acceleration at time \(t>0\) is given by \(a(t)=\cos t .\) At time \(t=0,\) its position is \(x=3\) (a) Find the velocity and position functions for the particle. (b) Find the values of \(t\) for which the particle is at rest.

Midpoint Rule Explain why the Midpoint Rule almost always results in a better area approximation in comparison to the endpoint method.

Acceleration The maker of an automobile advertises that it takes 13 seconds to accelerate from 25 kilometers per hour to 80 kilometers per hour. Assume the acceleration is constant. (a) Find the acceleration in meters per second per seconds. (b) Find the distance the car travels during the 13 seconds.

Acceleration Assume that a fully loaded plane starting from rest has a constant acceleration while moving down a runway. The plane requires 0.7 mile of runway and a speed of 160 miles per hour in order to lift off. What is the plane's acceleration?

Even and Odd Functions In Exercises 73-76, evaluate the integral using the properties of even and odd functions as an aid. $$\int_{-\pi / 2}^{\pi / 2} \sin x \cos x d x$$

Even and Odd Functions In Exercises 73-76, evaluate the integral using the properties of even and odd functions as an aid. $$\int_{-\pi / 2}^{\pi / 2} \sin ^{2} x \cos x d x$$

Step Function Evaluate, if possible, the integral $$\int_{0}^{2}[\sqrt{x}] d x$$

Proof Prove each formula by mathematical induction. (You may need to review the method of proof by induction from a precalculus text.) (a) \(\sum_{i=1}^{n} 2 i=n(n+1) \quad\) (b) \(\sum_{i=1}^{n} i^{3}=\frac{n^{2}(n+1)^{2}}{4}\)

Using a Riemann Sum Determine $$\lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(1^{2}+2^{2}+3^{2}+\cdots+n^{2}\right)$$ by using an appropriate Riemann sum.