Final Exam - Math 221-01 Calc 1 Final Exam Please show all ...

Math 221-01 Calc 1 Final Exam

Please show all work when possible, use fractions instead of decimals when you can, and simplify
all answers when sensible. Circle this sentence for one bonus point. Remember that the purpose of

this test is to test your knowledge of the material, not your neighbors’.

Name:

Your score is out of 100 possible points.

1. (6 points) Identify the following functions as odd, even or neither. Justify your answer.
(a) f (x) = x4 − x2

(b) g(x) = 1 + x2 + x3

(c) h(x) = √ + x2 )
3 x(1

Page 1 of 10

Math 221-01 Calc 1 Final Exam

2. (12 points) Evaluate the following limits, justifying your answer:
x5 + 5x + 6

(a) lim
x→∞ −4x5 + 3x4 + 20

x2 − 4
(b) lim

x→2 x2 − 5x + 6

(c) lim √ x − 1
x→1 x2 + 3 − 2

(d) lim √x2 − 9
x→−∞ x6 + 7

Page 2 of 10

Math 221-01 Calc 1 Final Exam

3. (9 points) For the following functions identify them as continuous or state the type of discontinuity
for each discontinuity. Justify your answers.
x3 − 2x + 1
(a) f (x) = x2 + 9

x2 + 3x + 2
(b) g(x) = x2 − 1

x2 − 4 if x < 2
(c) h(x) = x − 1 if x ≥ 2

Page 3 of 10

Math 221-01 Calc 1 Final Exam

4. (6 points) Using the Limit Definition of Derivative find f (x) for f (x) = x2 − 3.

√
5. (3 points) Write an equation for the line tangent to g(x) = x2 + 1 when x = 1.

Page 4 of 10

Math 221-01 Calc 1 Final Exam

6. (10 points) For each of the following functions, find y :
(a) y = 10x8 − 3x + 5

(b) y = ln(2 − x)

(c) y = ex

(d) y2 − yx2 = 4

x
(e) y =

x2 + 1

Page 5 of 10

Math 221-01 Calc 1 Final Exam

7. (10 points) Evaluate each of the following integrals or state why they don’t exist:
2x

(a) dx
x2 + 9

√
(b) x x − 1 dx

π

(c) 4 sec2(θ) dθ

0

(d) 3 √ 1 dt
0 t−2

1

(e) x99 dx

0

Page 6 of 10

Math 221-01 Calc 1 Final Exam

√
8. (4 points) For the function f (x) = x over the interval [1, 4], show that f (x) satisfies the conditions

for the Mean Value Theorem, and find the value c prescribed by the theorem.

x
9. (4 points) For the function f (x) = x2 + 1, find the absolute maximum and absolute minimum over

the interval [0, 2].

Page 7 of 10

Math 221-01 Calc 1 Final Exam

10. (8 points) A light is at the top of a pole 80 ft high. A ball is dropped at the same height from a point
20 ft from the light. Assuming that the ball falls according to s = 80 − 16t2, how fast is the shadow

of the ball moving along the ground 1 second later?

11. (4 points) For the function g(x) = x3 + 5x − 9, show that g(x) has only one root over the interval
[0, 2].

Page 8 of 10

Math 221-01 Calc 1 Final Exam

12. (12 points) For the function f (x) = x4 − 6x2:
(a) Find the intervals of increase or decrease.

(b) Find the local maximum and minimum values.

(c) Find the intervals of concavity and the inflection points.

(d) Sketch the graph of f (x):

Page 9 of 10