[12 1. pts.] In 1910 the population of Mars was about 12 million and growing at 3.5% annually. (a) Write an expression for the function f(t) that gives the population of Mars, in millions, t years after 1910. (b) Find the rate of growth, in people/year, of the population of Mars in 1915.
2. The dierentiable function g has the following properties: g(2) = 5, g (2) = 3, g(0) = 2, g (0) = 4. Show your work below. (a) If f(x) = ln(g(x)), nd f (2). 3. Find the slope of the curve xy + y2 = 1 at the point (0,1). 4. Consider the table of values for the function f: x 0 20 40 60 80 100 f(x) 1.2 2.8 4.0 3.2 5.1 5.2
100
Give upper and lower estimates to the value of the integral
c
f(x) dx. Show your work.
0
5. Find the number c for which
1
6 dx = 2. x2
1
6. Evaluate the indenite integral (1 + sin x 2ex ) dx. 7. Use the tangent approximation to estimate (7.88) 3 . 1 8. The function f is dened on (, 0) (0, ) and has derivative f (x) = x x . (a) Find the intervals on which f is increasing and those on which f is decreasing. (b) Find the intervals on which f is concave up and those on which f is concave down. 8 9. Determine the equation of the line tangent to the curve y3 + y = xy + x + 2 at the point ( 3 , 2). 10. The point A moves along the x-axis at constant rate of 4 ft/sec. while a second point B moves along the y-axis at a constant rate of 2 ft/sec. Find how fast the distance between the points is changing when A is (3,0) and B is (0,4). 11. A serindiptogon is a solid gure whose surface area S and volume V, determined by the radius r and 2 the height h, are S = 6 hr + h2 3h and V = rh2 . You want to build one of these things with a xed volume V = 100. Find the dimensions r and h that minimize the surface area. 12. The acceleration of gravity is -32 ft/sec. A stone is dropped from a height 100ft. of What is its velocity when it hits the ground? 5 13. Find the antiderivative ( 7 + sec2 x 3 sin x)dx. x 14. An elevator moves with velocity v(t) = 2t3 3t2 36t + 1 at time t. (a) Find an expression for the acceleration a(t) at time t. (b) At what times between t = 0 and t = 4 is the velocity increasing? (c) What is the change in position of the elevator between times t = 0 and t = 4? (d) What is the total distance traveled by the elevator between times t = 0 and t = 4?
cos x
15. Find the derivative h (x) of the function h(x) =
1
t4 dt.
16. Find the area of the region bounded by the curve x = 0, x = 2, y = (x 1)3 + 1 and the x-axis.
x
17. Let g(x) =
1
f(t)dt where the graph of the function f is illustrated below.
Something p.w. linear so the regions divide nicely into rectangles and triangles (a) Find g(1)= (b) g(2)= (c) g(6)= (d) On what interval is g increasing? (e) Where does g have a maximum? (f ) Find g(2.5)= (g) g(4)= 18. (5 pts) Dene precisely what it means for a function to be continuous at x = a. 2x 5 if x 1 19. (14 pts) Let f(x) = x2 if ...