The vertical ends of a water trough are isosceles triangles $5$ feet across the top and $5$ feet deep. Calculate the force on an end when the trough is full of water.

Approximate the sum of the series correct to four decimal places. ∑∞ n=1 (-1)^n / (2n)!

(a) What is the buoyant force on 0.90 kg of ice floating freely in liquid water? (b) What is the buoyant force on 0.90 kg of ice held completely submerged under water?

Find the volume of the solid obtained by rotating the region enclosed by the curves about the given axis.

y=2x, y=0, x=8; x-axis

A fourth-degree polynomial function has real zeros -2, 0, 1, and 5. Find two different polynomial functions, one with a positive leading coefficient and one with a negative leading coefficient, that could be g. How many different polynomial functions are possible for g?

Determine whether or not the vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf{F}=\nabla f$.

$$\mathbf{F}(x, y, z)=\langle y z, x z+y, x y-x\rangle$$

A singing contest eliminates contestants after each round. A rational exponent (n - 1)/n, where n represents the round of the competition is used to determine the number of contestants that will go to the next round. The number of contestants in Round 2 is 243. How many contestants will be in Round 5?

Sketch the graph of the function

$$f(x)=\frac{x^{2}+2 x-3}{x-1}.$$

Analyse its behaviour near its non-permissible value.

Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires a force of 50 N to be stretched 0.1 m from its equilibrium position. b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position.

Calculate the work required to stretch the following springs 1.25 m from their equilibrium positions. Assume Hooke's law is obeyed. a. A spring that requires 100 J of work to be stretched 0.5 m from its equilibrium position. b. A spring that requires a force of 250 N to be stretched 0.5 from its equilibrium position.

Calculate the mass and centroid (center of mass) of the following thin plate, assuming constant density.

The region bounded by $y=1-|x|$ and the $x$-axis

Calculate the mass and centroid (center of mass) of the following thin plate, assuming constant density.

The region bounded by $y=\ln x$, the $x$-axis, and $x=e$

The sun is 20° above the horizon. Find the length of a shadow cast by a park statue that is 12 feet tall.

Let f be the function whose graph is shown. On which of the following intervals, if any, is f continuous?

$[2,3]$

For each interval on which f is not continuous, indicate which conditions for the continuity of f do not hold.