Math 151 Exam 2 Review Wed March 25, 7-9 pm Blocker 102

1. Evaluate each limit as a number, infinity, or -infinity or state DN E.

b) c) d)

e) f) g)

2. a) Show that the tangent to a circle is perpendicular to a radial line through the center and the point of tangency using implicit differentiation on the equation for the unit circle.

b) Is the same true for the ellipse ?

3. A curve is described by . Use implicit differentiation to find y'.

4. A curve has equation Find the tangent line to the curve at the point (1, 0).

5. a) Find the equation of the tangent line to this curve at the point (0,1 )

5b) Find both tangent lines.

6. Find the 127th derivative with respect to x of f(x)=cosx.

7. Show that solves the differential equation y" + 2y' + y = 0

8. For what values of a does solve the differential equation y" = y' + y ?

9. where p(x) is differentiable three times. Find the third derivative of f(x) with respect to x.

10. A tank is in the shape of an inverted cone having height 16 ft. and top radius of 4 ft. Water is filling the tank at the rate of 2 cubic feet per minute. How fast is the depth of the water rising when the water is 5 feet deep?

11. Car A is traveling due east toward an intersection at 40 mph. Car B is traveling due north toward the same intersection at 30 mph. How fast is the distance between them decreasing when car A is 300 feet away and car B is 400 feet away from the intersection. Note, the conversion from feet to miles will cancel out.

12. h(x) = f(g(x)). L1(x)= 5x-3 is the linear approximation to g(x) near x=1. L2(u) = 4u+5 is the linear approximation to f(u) near u=2. Find the linear approximation to h(x) near x=1.

13. Use linear approximation to approximate .

14. If Newton's method is used to approximate using 2 as the first guess, find .

15. Find the inverse function to f(x)= . Find the asymptotes of f and the asymptotes of the inverse.

16. Find the derivative of (u) at any u.

17. The tangent line to a function, f, at x=1 is y = -3x + 5. f is one to one near x = 1. Find the tangent line to the inverse function at x=f(1).

18. Find the domain and range of each function:

a)

19. Solve each equation for x.

a) b) c)