A baseball pitcher throws a baseball with an initial speed of 91 feet per second at an angle of 5 degree to the horizontal. The ball leaves the pitcher's hand at a height of 15 feet. Write the parametric equations that describe the motion of the ball as a function of time.

a) Write a set of parametric equations that model the baseball.

b) What is the max height of the ball?

c) When does the ball hit the ground?

d) How far away from the pitcher does the ball hit the ground?

e) Will the batter hit the ball if the batter is 60 feet away and swings the bat at a height of 3 feet? Why or why not?

A business owner needs to run a gas line from his business to a gas main as shown in the accompanying diagram. The main is 30-ft down the 12-ft wide driveway and on the opposite side. A plumber charges $4 per foot alongside the driveway and $5 per foot for underneath the driveway.

a) What will be the cost if the plumber runs the gas line entirely under the driveway along the diagonal of the 30-ft by 12-ft rectangle? b) What will be the cost if the plumber runs the gas line 30-ft alongside the driveway and then 12-ft straight across? c) The plumber claims that he can do the job for $160 by going alongside the driveway for some distance and then going under the drive diagonally to the terminal. Find x, the distance alongside the driveway. d) Write the cost as a function of x and sketch the graph of the function. e) Use the minimum feature of a graphing calculator to find the approximate value for x that will minimize the cost. f) What is the minimum cost (to the nearest cent) for which the job can be done?

1. ʃ11-3x/(x^2+2x-3) dx

2. ʃ4x-16/(x^2-2x -3 )dx

3. ʃ(2x^2-9x-35)/(x+1)(x-2)(x+3) dx

Evaluate the following integral ∫ ∫ R 4x + y (3x − y) ln(3x − y) dA where R is the region bounded by the graphs of  y  =  3x − e7,  y  =  3x − e5,  y  =  −4x + 8,  and  y  =  −4x + 5. Use the change of variables  u  =  3x − y,  v  =  4x + y.

QA: (inspired by a problem from Stewart’s textbook) Suppose that there is some function f whose _derivative_ is f ’ =sin(x)/x, with f ‘(0)=1 by definition rather than DNE. Draw that, on the interval [-4pi,+4pi].

(i) On what intervals is the original f increasing? Decreasing? Indicate the intervals on the graph as well as writing them in interval notation like [0,pi]

(ii) At what x values does f have a local max? A local min? Indicate them on the graph as well as writing them out like: maxes at …. ; mins at ….

(iii) On what intervals is f CD? CU? Indicate the intervals on the graph as well as writing them in interval notation.

(iv) At what x values does f have an inflection point? Indicate them on the graph as well as writing them out like: IP at ...

(v) Sketch a graph of f, starting at f(0)=0.

Solve 4 questions of quiz. each of them gives 0.25 point

1. Show that the following sets of elements in R³ form subspaces. (a). The set of all (x, y, z) such that x − 2y + z = 0. (b). The set of all (x, y, z) such that x = 3z and y = z.

2. (a). Let U = {(x, y) ∈ R² : 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1} and W = {(x, y) ∈ R² : x 2 + y 2 ≤ 1}. Are these sets subspaces of R^2? (b). Find the sum U + W.

3. If U and W are subspaces of a vector space V, show that U + W is a subspace

4.Show that functions f(t) = t and g(t) = 1/t defined for t > 0 are linearly independent.

The height of a rock thrown upwards on the planet Mars with an initial velocity of 60 m/s is 2 approximated by h(?) = 60? − 1.86? . Fill in the average velocity over each time intervals shown in the table. Show all work for each calculation. Then approximate the instantaneous velocity of the rock after 10 seconds using the table of values.(?h?? ??? ???? ???? ???h ???????????; ?? ??? ????? ??? ???????. )

find the equation of tangent line to the curve r=1+3sinθ when θ=π/3

Let f(x) = cosh(x) and g(x) = sinh(x), a = 0 and b = 1.

A) Find the volume of the solid with base on the xy plane, bounded by the region above, whose cross-sections perpendicular to the x axis are squares.

B) Find the volume of the solid formed if the region above is rotated about the line y = 4.

C) Find the volume of the solid formed if the region above is rotated about the line x = 2.

Sketch the graph of the curve ? = 1 + sin ? for 0 ≤ ? ≤ 2? ? b) Find the slope of the tangent line to this curve at ? = 4 . c) Find the polar coordinates of the points on this curve where the tangent line is horizontal.

1. Find answers that meet the first-order optimization conditions, identify the shape of the function graph to determine whether these answers are maximum value or minimal value.

(a) f(x) = e^x + e^-x

(b) f(x) = x^3/(x+1)

(c) f(x) = x/lnx

2. Find answers that meet the second-order optimization conditions, and determine whether these answers are maximum value or minimal value.

(a) f(x) = e^x + e^-x

(b) f(x) = x^3/(x+1)

(c) f(x) = x/lnx

3. Determine whether each function is concave or convex. If the entire function is not concave or convex, look for concave and convex sections.

For the following u(x, y), show that it is harmonic and then find a corresponding v(x, y) such that f(z)=u+iv is analytic.

u(x, y)=(x^2-y^2) cos(y)e^x-2xysin(y)ex

The position of a particle moving along a line (measured in meters) is s(t) where t is measured in seconds. Answer all parts, include units in your answers.

s(t)=2t^3 +6t^2 −48t+7 −10<t<10

(a) Find the velocity function.

(b) Find all times at which the particle is at rest.

(c) On what interval is the particle moving to the right?

(d) Is the particle slowing down or speeding up at t = −1 seconds?

In the new Mission Impossible movie, Ethan Hunt must save the world from an evil genius who is going to release a computer bug that will shut down the whole power grid. In an attempt to creative and critically think through how to stop this evil genius, Ethan Hunt finds himself on the London Eye, Great Britain's largest Ferris Wheel. The Eye is centered at (0,75) with a radius of 60 meters. It makes one revolution every 24 minutes. The London Eye loads from bottom.

a) Write a set of parametric equations for the London Eye.

b) If it Ethan Hunt 60 minutes on the London Eye to figure out a plan, how far did he travel?

c) After Ethan Hunt figures out his plan, Agent Luther Stickell joins him on the Ferris Wheel. When is Agent Luther Stickell at a height of 100 meters?

Find the surface area of revolution about the x-axis of y = 5 sin ( 5 x ) over the interval 0 ≤ x ≤ π/5