Math calculus
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1. Let ?? ??

A) Find and simplify ?? ?? and ???? ??

B) Solve ?? ?? 0 and ?? ?? 0. Also find the y-coordinates of those x- values.

C) Find an equation for the secant line to ?? ?? at x = 2.

D) Find a point, x, in the interval (0, 2) with ?? ?? .25

2. Let ?? ?? ?

2 Find the x and y coordinates of the place(s) where T(x) has horizontal

tangents.

3. Let f(x) = sin(2 x) + 2 sin(x). Find a formula for all of the x- and y- coordinates of the places where f(x) has horizontal tangents.

4. Find the x-intercept of the tangent line to f(x) = sec2(2x) at x = ?/6.

Homework #5 two extended word problems, so worth a total of 40 pts.

1. A projectile is launched vertically from a launch pad with an unknown initial velocity, V0. An unknown time, T, later, an observer is in a pickup which is traveling in such a way that the angle of elevation to the projectile is 22° and increasing at 2° per second. When the pickup is 750 m away from the launch pad, it is moving away from the launch pad at 25 m/sec. A) Find the initial velocity, V0. B) Find the time, T, since launch. C) Find the projectile’s maximum height. D) Find the time until the projectile hits the ground again. Calculate all of these to 3 decimal places. Hint: . First solve the Related Rate problem to find the projectile’s velocity and height. . Then use the fact that h(t) = -4.9t2 + V0 t , and that v t h‘ t . 2. A woman is on an island 4 miles from a straight shoreline, and wants to reach a point 5 miles along the shoreline. She can run at 10 miles per hour. She also has a rowboat. A) Find the minimum speed she needs to be able to row, so that she would be best off rowing directly to her destination. B) Show that, no matter how slowly she rows (at any speed greater than zero), she is never best off rowing directly to the shore there’s always some point slightly down the shoreline that is optimal.