applied clac study guide help

Applied Calculus (Math 026) Fall 2017 Final Study Guide Howard University Department of Mathematics

Dr. G. D. Mc Neal

Name:

Show all your work in the bluebook provided. No work, No credit. Best Wishes!

PART I: Do all of the problems in part I.

1.15points Evaluate the following limits, or if any does not exist, classify it as −∞ or +∞, if appropriate:

(a) lim x→0

(2 − 1

x5 )

(b) lim x→−∞

−x7 + 3x4 + 5x2 − 2x + 8 1 + 5x2 + x5

(c) lim x→1

x− 1 √ x2 + 3 − 2

2.20points Let g(x) be defined as follows:

g(x) =

 

x2 − 6x + 9 x− 3

if x > 3

x3 + 2x− 33 if x ≤ 3

(a) State a precise definition of what it means for an arbitrary function f(x) to be continuous at x = a.

(b) Evaluate each of the following or briefly explain why it does not exist: (i) g(3), (ii) lim

x→3− g(x), (iii) lim

x→3+ g(x).

(c) Briefly explain why lim x→−4

g(x) does or does not exist.

(d) Use the definition of continuity to decide if g(x) is continuous at x = 2. Justify your answer.

3.10points Using the definition of the derivative, f ′(x) = lim h→0

f(x + h) −f(x) h

, compute f ′(x), and

simplify given f(x) =

√ 1 + 5x.

4.15points Differentiate each of the given functions and simplify:

(a) f(x) = ln

3 √ x2

x4 .

(b) g(t) = √

ln t + t5.

(c) p(t) = ex + e−x

ex −e−x .

5.20points For the function f(x) = e2x

x2

(a) Find the rate of change of f when x = 1.

(b) Write an equation of a line tangent to the graph of f at x = 1.

6.20points For the function f(z) = 2z3 + 6z2 + 6z + 5

(a) Find the intervals where f is increasing and decreasing.

(b) Classify the critical numbers as relative maximum and minimum.

(c) Find f ′′(z).

(d) Find the intervals where the graph of f is concave upward and concave downward.

(e) Find the inflection point(s), if any.

7.20points Evaluate the following integrals:

(a)

∫ 9 1

(√ x−

4 √ x

) dx

(b)

∫ (x3 + x)

√ x4 + 2x2 + 1 dx

(c)

∫ ( ex + 1

)3 ex dx

(d)

∫ 4 2

(t− 1)(t + 3)8 dt

8.20points Use Integration by parts to find the given integrals:

(a)

∫ x √ x + 5 dx

(b)

∫ 1 0

t (e−2t + e−t) dt

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PART II: Do ONLY THREE of the problems in part II.

9.20points By using logarithmic differentiation, find the derivative of

f(x) = (x + 1)3 (6 −x)2 5 √

3x + 5.

10.20points Use implicit differentiation to find dy/dx if;

(1 − 2xy3)5 = 2x + 4y

11.20points Find the absolute maximum and absolute minimum (if any) of the curve

f(x) = 2x3 − 3x2 − 12x + 5, −3 ≤ x ≤ 3.

12.20points Let R be the region bounded by the curves: y = x2 + 2, y = x− 3, x = −3, and x = 2.

(a) Sketch the region R bounded by the given curves.

(b) Find the area of the region R.

13.20points By using integration by parts find

∫ x ln 2x dx.

14.20points Compute all second order partial derivatives of

f(u, v) = uv3 + 5uv2 + 2u + 8.

That is, compute fu, fv, fuu, fvv, fuv, fvu.

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