Math Homework

Math 21A Name: Summer I 2018 Practice Final 8/2/2018 Time Limit: 100 Minutes Instructor

1. Do not turn the page until told to do so. 2. It is a violation of the university honor code to, in any way assist another person in the completion of this exam. Please keep your own work covered up as much as possible during the exam so that others will not be tempted or distracted. Thank you for your cooperation. Violations can result in expulsion from the university. 3. You may use 1 page (Standard 8.5 × 11 inch) handwritten front and back crib sheet, but no calculator, and no classmates may be used as resources for this exam. 4. Read directions to each problem carefully. Show all work and simplify your answers for full credit. In most cases, a correct answer with no supporting work will receive little to no credit. What you write down and how you write it are the most important means of you scoring well on this exam. Neatness and organization are also important. 5. You may NOT use shortcuts for finding limits to infinity. Show enough work for full credit. 6. You will be graded on proper use of mathematical notation. 7. You have until 5:50 PM to finish the exam.

Grade Table (Instructor use only)

Question Points Score

1 10

2 10

3 10

4 10

5 10

6 10

7 10

8 10

9 10

10 10

11 0

Total: 100

Math 21A Practice Final 8/2/2018

1. (10 points) Determine the following limits:

lim x→∞

x + sin x + 2 √ x

x + sin x

lim x→∞

x2/3 + x−1

x2/3 + sin2 x

Math 21A Practice Final 8/2/2018

2. (10 points) If limx→0+ g(x) = L1 and limx→0− g(x) = L2, find

(a) lim x→0+

g(x3 −x)

(b) lim x→0−

g(x3 −x)

(c) lim x→0+

g(x2 −x4)

(d) lim x→0−

g(x2 −x4)

Briefly reason why for each part.

Math 21A Practice Final 8/2/2018

3. (10 points) Find the equations for the lines that are tangent and normal to the curve at the given point:

x3/2 + 2y3/2 = 17, (1, 4)

Math 21A Practice Final 8/2/2018

4. (10 points) Graph the function

f(x) =

{ x, 1 ≤ x < 0 tan x 0 ≤ x ≤ π

4

Is f continuous at x = 0? Is f differentiable at x = 0? Justify your answer.

Math 21A Practice Final 8/2/2018

5. (10 points) Show that f(x) = x3 + 10 has exactly one root.

Math 21A Practice Final 8/2/2018

6. (10 points) Let

f(x) = x

x + 1

g(x) = x5 + 4x

Show that f(x) is increases on every open interval in its domain. Also argue why g(x) can’t have any local extrema (Your argument must be correct and concise).

Math 21A Practice Final 8/2/2018

7. (10 points) Graph y = x √

4 −x2. This includes finding where y increases, decreases, local extrema (including absolute extrema), concavity, point of inflection(s), any asymptotes, intercepts (both x and y). You solution MUST be clear and UNCLEAR solutions will typically receive little to zero points!

Math 21A Practice Final 8/2/2018

8. (10 points) Compute the following limits:

lim x→∞

( 1 +

4

x

)−3x lim x→0

2−sinx − 1 ex − 1

lim x→0

sin mx

sin nx

Math 21A Practice Final 8/2/2018

9. (10 points) An isoceles triangle has its vertex at the origin at the origin and its base parallel to the x-axis with the vertices above the axis on the curve y = 31 − x2. Find the largest area the triangle can have.

Math 21A Practice Final 8/2/2018

10. (10 points) Given tan x crosses the line y = 2x between x = 0 and x = π 2 . Solve for the

value of x where this graph intersect (Numerically).

Math 21A Practice Final 8/2/2018

11. (0 points) Extra Credit???? Not going to ruin a good surprise ;)