calculus

Final Exam

Instructor: Holmes

1. Given f(x) = x2 ln 5x,

a. Find the domain, intercepts, and when the function is above and below the x-axis. Determine if the function has any symmetry. Find any asymptotes.

b. Find the intervals where the function is increasing and decreasing. Locate and classify all extrema.

c. Find the intervals where the function is concave up and concave down. Locate any inflection points.

d. Sketch the graph of the function. Label all significant points. State the range.

2. Find the indefinite integrals.

a. ∫ e−2xdx

b. ∫

sin(πx)dx

c. ∫

sec 1 2 θ tan 1

2 θdθ

d. ∫

cos πx 3 dx

3. Evaluate each limit.

a. lim x→0+

(1 + x)1/x

b. lim x→0

sin 3x cos 5x

3x

4. Find the derivative of each.

a. y = (x2 + 1)(x + 3)1/2

x− 1 b. y = (2 + sin 3x)π

c. y = xtan −1 x

d. y = 3 (5x2 + sin 2x)3/2

e. y = x−e + e−x

f. y = (x2 + 5)2−7x 2

1

4. Determine if the function is continuous at the given point. Graph each function.

a. f(x) =

{ 3x− 2 if x ≤ 3, 5 −x if x > 3

at a = 3

b. f(x) =

{ x2 − 4 x− 2 if x 6= 2,

4 if x = 2 at a = 2

5. Find an equation of the tangent line to the graph of the function at the prescribed point.

a. x2 + y2 = 5x + 4y at (5, 4)

6. Determine the values of A and B so that the function f is continuous at every real number and sketch the graph of the resulting function

f(x) =

 

3x if x ≤ 2, Ax + B if 2 < x < 5

−6x if x ≥ 5

7. Let f(x) = 2 √ x. Show that f satisfies the Hypotheses of the Mean Value theorem on

[1, 4] and find all real numbers c guaranteed by the theorem.

8. A farmer with 400m of fencing wants to enclose a rectangular plot that borders a straight river. If the farmer does not vince the side along the river, what is the largest rectangular area that can be enclosed?

9. A 22-foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 ft/sec, how fast is the top of the ladder moving down when the foot of is 15 feet from the wall?

10. Sketch the region and find its area. The region bound by y = 144 36 + x2

and y = 1

11. Find the solution to the following initial value problem.

g′(x) = 4x

( x3 −

1

4

) ; g(1) = 1

2