MTH 263 Test 1 Study Guide Fall 2020 Name

Mth 263 Test 1 Study Guide Fall 2020name

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) = x³, and h(x) = 3x+ 6, find h(g(f(x))).

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

3) y = (-3x + 18)⁸

4) Let g(x) = x + 6. Find a function y = f(x) so that (f g)(x) = 4x + Graph the function.

5) y = x + 4

6) 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.

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

8) Solve the problem. 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.

9) 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?

10) Provide an appropriate response.

11) Consider the graph of f(x) = 9 - x², 0

12) Express as a single logarithm and, if possible, simplify. ln (32x + ln 4)

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

14) Use the graph to find a > 0 such that for all x, 0

15) y = x + 3

16) y = x - 2

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

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

19) Prove the limit statement.

20) lim_x→7 (x² - 49) / (x - 7) =

21) Solve the problem. 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?

22) 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 0, there exists a value of x for which f(x) - L

23) Find the intervals on which the function is continuous.

24) y = 3x + 7 - 3x

25) 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

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

27) lim_x→∞ (16x² - 3 + 9x)

28) lim_x→25 (x² + x - 3) / ((x - 11)(x + 6))

Paper For Above instruction

In this comprehensive review, we explore key concepts and problem-solving strategies in calculus, focusing on function compositions, limits, continuity, derivatives, and applications in real-world contexts. Each topic is carefully examined with detailed explanations and examples to enhance understanding and prepare for assessment in MTH 263.

Understanding Function Composition and Graphing

Function composition is a fundamental operation where one function is applied to the result of another. For instance, given functions f(x) = x, g(x) = x³, and h(x) = 3x + 6, we can find h(g(f(x))) by sequentially applying each function: f(x) first, then g, then h. This emphasizes the importance of understanding how functions interact and how to construct composite functions.

Graphing functions such as y = x + 4 or y = x² - 1 stretched horizontally by a factor of 4 provides visual intuition about transformations. Stretching horizontally by a factor alters the x-coordinates, effectively compressing or expanding the graph along the x-axis, which can be expressed as f(x) = g(bx) where b adjusts the stretch factor.

Application of Exponential Decay and Half-life

The decay of radioactive material is modeled by exponential functions like A(t) = 608e^-0.019t. For example, after 50 years, the remaining amount can be calculated by substituting t = 50, illustrating the practical use of exponential decay formulas. Similarly, understanding half-life helps determine how long it takes for a substance to decay to a certain percentage of its original amount, using half-life formulas derived from exponential decay models.

Logarithmic Functions and Law-of-Logs

Logarithmic expressions such as ln(32x + ln 4) can be simplified using properties of logarithms: ln(ab) = ln a + ln b, and ln(a/b) = ln a - ln b. These properties allow consolidating multiple logarithms into a single expression, facilitating easier manipulation and solving of equations involving logs.

Limits, Continuity, and Derivatives

Fundamental limit concepts, such as lim_x→c (f(x) - L) = 0 (or the epsilon-delta definition), are crucial for understanding the behavior of functions near specific points. For example, to evaluate the limit of a rational function as x approaches a point where the function appears to be undefined, algebraic manipulation like dividing numerator and denominator by the highest power of x in the denominator is often used to simplify the expression. The concept of continuity, characterized by the limit existing and matching the function's value at a point, is essential for analyzing function behavior.

Practical Applications: Ohm's Law and Tolerance Intervals

Ohm's Law, V = RI, links voltage, resistance, and current, allowing us to determine appropriate resistance ranges for specified voltage and current tolerances. For example, if V = 12 volts and I = 5 ± 0.1 amperes, solving for R provides an interval for resistance that maintains the current within acceptable bounds, illustrating the relevance of limits and inequalities in engineering contexts.

Identifying and Correcting Limit Misconceptions

Common misconceptions about limits include misunderstanding the epsilon-delta definition or misinterpreting the behavior of functions near specific points. Accurate understanding requires careful application of inequalities and formal definitions to establish limit existence and value.

Continuity and Extensions at Boundary Points

Graphing tools help determine whether a function can be extended to be continuous at points where the original function may be undefined or discontinuous. For example, if evaluating the limit of a function at zero suggests a particular value, one can define the function at that point accordingly to achieve continuity, provided the limit exists from the right or from the left.

Advanced Limit Techniques

Limits involving indeterminate forms, such as 0/0 or ∞/∞, often necessitate algebraic techniques like factoring, rationalizing, or applying L'Hôpital's Rule (if appropriate). For instance, evaluating lim_x→7 (x² - 49) / (x - 7) involves recognizing the numerator as a difference of squares and factoring accordingly.

Conclusion

This review emphasizes the interconnectedness of calculus concepts, from basic function properties to complex applications in science and engineering. Mastery of these topics requires understanding both the theoretical foundations and the practical problem-solving techniques that allow for effective analysis of real-world systems.

References

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