Find the curvature at the given point on the polar curve given.

$$r=1-2 \sin \theta, \text { where } \theta=\frac{\pi}{4}$$

A child slides down a hill on a toboggan with an acceleration of $1.6 \mathrm{~m} / \mathrm{s}^2$. If she starts at rest, how far has she traveled in $1.0 \mathrm{~s}$ ?

Determine which of the following limits exist. Compute the limits that exist.

$$\lim _{x \rightarrow 4} \frac{x^2-16}{1-x}$$

At the beginning of the mentioned act, what arrangement between Ferdinand and Miranda does Prospero approve?

Prove that any integer n can be written as a fraction.

Find the values of a and b for the curve x²y+ay²=b if the point (1, 1) is on its graph and the tangent line at (1, 1) has the equation 4x+3y=7.

Explain the importance of

caliph

Shi'a

Sunni

In Exercise, uses $\kappa ( t ) = \frac { \left| x ^ { \prime } y ^ { \prime \prime } - y ^ { \prime } x ^ { \prime \prime \prime } \right| } { \left( x ^ { \prime 2 } + y ^ { \prime 2 } \right) ^ { 3 / 2 } }$ and $\kappa ( x ) = \frac { \left| d ^ { 2 } y / d x ^ { 2 } \right| } { \left[ 1 + ( d y / d x ) ^ { 2 } \right] ^ { 3 / 2 } }$, to find the normal scalar component of acceleration as a function of $x$ .
(a) A paricle moves along the parabola $y = x ^ { 2 }$ with a constant speed of 3 units per second.

Find

$$f(f(x))$$

where $f(x)=5x-1$.

Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots.

$$5x^2+2x+4=0$$

Find parametric equations for the tangent line to the curve of intersection of the cylinders

$$x ^ { 2 } + z ^ { 2 } = 25$$

and

$$y ^ { 2 } + z ^ { 2 } = 25$$

at the point (3, -3, 4).

Find the solution of the differential equation that satisfies the given conditions.

$$y^{\prime \prime \prime}+y^{\prime \prime}=4 e^{-2 t}, \quad y(0)=2, \quad \lim _{t \rightarrow \infty} y(t)=1$$

Analyze the importance of setting precedents during Washington's presidency.

(i) Find $\int\left(\frac{1}{x^{2}}-\frac{1}{\sqrt{x}}\right) d x$. (ii) A curve passes through the point (1, 0) and is such that $\frac{d y}{d x}=\frac{1}{x^{2}}-\frac{1}{\sqrt{x}}$. Find the equation of the curve.