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MTH 263 Test 1 Study Guide Fall 2020

Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Solve the problem.

1) If f(x) = x, g(x) = x 3

, and h(x) = 3x+ 6, find h(g(f(x))). 1)

Express the given function as a composite of functions f and g such that y = f(g(x)). 2) y = (-3x + 18)8 2)

Solve the problem. 3) Let g(x) = x + 6. Find a function y = f(x) so that (f g)(x) = 4x + 24 3)

1

Graph the function. 4) y = x + 4 4)

The problem tells by what factor and direction the graph of the given function is to be stretched or compressed. Give an equation for the stretched or compressed graph.

5) y = x2 - 1 stretched horizontally by a factor of 4 5)

Solve the problem. 6) The decay of 608 mg of an isotope is given by A(t)= 608e-0.019t, where t is time in years.

Find the amount left after 50 years. 6)

2

7) In the formula A = Iekt, A is the amount of radioactive material remaining from an initial amount I at a given time t and k is a negative constant determined by the nature of the material. A certain radioactive isotope has a half-life of approximately 1450 years. How many years would be required for a given amount of this isotope to decay to 60% of that amount?

7)

Provide an appropriate response.

8) Consider the graph of f(x) = 9 - x2, 0 x 1. What symmetry does the graph have? Is f its own inverse?

8)

Express as a single logarithm and, if possible, simplify. 9) ln (32x + 16) - 2 ln 4 9)

Find the average rate of change of the function over the given interval. 10) y = x2 + 9x, [1, 8] 10)

3

Use the graph to find a > 0 such that for all x, 0 < x - c < f(x) - L < . 11)

y = x + 3 5.2

5

4.8

1.8 2 2.2

NOT TO SCALE

f(x) = x + 3 c = 2 L = 5

= 0.2

11)

12)

y = x - 2

1.25 1

0.75

2.5625 3 3.5625

NOT TO SCALE

f(x) = x - 2 c = 3 L = 1

= 1 4

12)

4

A function f(x), a point c, the limit of f(x) as x approaches c, and a positive number is given. Find a number > 0 such that for all x, 0 < x - c < f(x) - L < .

13) f(x) = 4x + 1, L = 13, c = 3, and = 0.01 13)

Prove the limit statement

14) lim x 7

x2 - 49 x - 7

= 14 14)

Solve the problem. 15) Ohm's Law for electrical circuits is stated V = RI, where V is a constant voltage, R is the

resistance in ohms and I is the current in amperes. Your firm has been asked to supply the resistors for a circuit in which V will be 12 volts and I is to be 5 ± 0.1 amperes. In what interval does R have to lie for I to be within 0.1 amps of the target value I0 = 5?

15)

Provide an appropriate response. 16) Identify the incorrect statements about limits.

I. The number L is the limit of f(x) as x approaches c if f(x) gets closer to L as x approaches x0.

II. The number L is the limit of f(x) as x approaches c if, for any > 0, there corresponds a > 0 such that f(x) - L < whenever 0 < x - c < .

III. The number L is the limit of f(x) as x approaches c if, given any > 0, there exists a value of x for which f(x) - L < .

16)

5

Find the intervals on which the function is continuous.

17) y = 3

x + 7 - 3x 17)

Provide an appropriate response. 18) Use a calculator to graph the function f to see whether it appears to have a continuous

extension to the origin. If it does, use Trace and Zoom to find a good candidate for the extended function's value at x = 0. If the function does not appear to have a continuous extension, can it be extended to be continuous at the origin from the right or from the left? If so, what do you think the extended function's value(s) should be?

f(x) = 6x - 1

x

18)

Divide numerator and denominator by the highest power of x in the denominator to find the limit.

19) lim x

16x2

3 + 9x2 19)

20) lim x

25x2 + x - 3 (x - 11)(x + 1)

20)

6

Answer Key Testname: MTH 263 TEST 1 STUDY GUIDE FALL 2020

1) x + 6 2) f(x) = x8, g(x) = -3x + 18 3) f(x) = 4x 4)

5) y = x2 16

- 1

6) 235 mg 7) 1069 yr 8) The graph of f is symmetric with respect to the line y = x. The function f is its own inverse because (f f)(x) = x. 9) ln (2x + 1)

10) 18 11) = 0.2 12) = 0.4375 13) = 0.0025 14) Let > 0 be given. Choose = . Then 0 < x - 7 < implies that

x2 - 49 x - 7

- 14 = (x - 7)(x + 7)

x - 7 - 14

= (x + 7) - 14 for x 7 = x - 7 < =

Thus, 0 < x - 7 < implies that x2 - 49 x - 7

- 14 <

15) 40 17

, 120 49

16) I and III 17) discontinuous only when x = -7 18) continuous extension exists at origin; f(0) 1.8145

19) 4 3

20) 5

7