umich math 11526 Calculus 1 Summary Notes [Chapter 1. FOUNDATION FOR CALCULUS: FUNCTIONS & LIMITS] 1.1 Functions and Change 1.2 Exponential Functions 1.3 New Functions from Old 1.4 Logarithmic Functions 1.5 Trigonometric Functions 1.6 Powers, Polynomials, and Rational Functions 1.7 Introduction to Limits and Continuity 1.8 Extending the Idea of a Limit [Chapter 2. KEY CONCEPT: THE DERIVATIVE] 2.1 How Do We Measure Speed? 2.2 The Derivat.. 2022. 12. 13. [Notes & HW Answers] 6.2 Constructing Antiderivatives Analytically [Prepwork 6.2] Q1. If 𝐹(𝑥) is an antiderivative of 𝑓(𝑥), what is the most general antiderivative? A: The most general antiderivative = F(x) + C Q2. Using the properties of antiderivatives in theorem 6.1 to find ∫(3cos(𝑥)+2sin(𝑥))𝑑𝑥 A: ∫(3cos(𝑥)+2sin(𝑥))𝑑𝑥= 3sin(x) - 2cos(x) + C Q3. Use the Fundamental Theorem to calculate the following integral exactly: A: ∫^(𝜋/3)_0 (2/cos^2𝜃) 𝑑𝜃= 2tan(𝜋/3) - 2t.. 2022. 12. 8. [Notes & HW Answers] 6.1 Antiderivatives Graphically and Numerically [Prepwork 6.1] Q1. Enter two different antiderivatives of 𝑓=2𝑥. A: Antiderivative 1 = x^2+1 Antiderivative 2 = x^2+2 Q2. Suppose that the graph of 𝑓′f′ is given below. At what 𝑥 value does 𝑓 cease being linear? 𝑥= 1 At what 𝑥 value does the maximum value of 𝑓 occur? 𝑥= 3 Q3. Suppose that 𝐹′(𝑡)=𝑡cos(𝑡) and 𝐹(0)=3. Use the data and method from example 5 in the text to estimate each of the followin.. 2022. 12. 8. [Notes & HW Answers] 5.4 Theorems about Definite Integral [Prepwork 5.4] Q1. For some function f, suppose that for some a∫^b_a f(x) dx=7 and ∫^c_a f(x) dx=9. Find each of the following. A: ∫^a_b f(x) dx= −7 ;∫^c_b f(x) dx= 2 Q2. Use the technique of example 2 in the text to evaluate the integral ∫^2_(−2) (|2x|+3) dx exactly. A: ∫^2_−2 (|2x|+3) dx = 20 Q3. Using a calculator to evaluate the appropriate integral, find the average value of P = f(t) = 2.03.. 2022. 12. 6. [Notes & HW Answers] 5.3 The Fundamental Theorem of Calculus [Prepwork 5.3] Q1. Around the beginning of the 1800’s, the population of the U.S. was growing at a rate of about 1.37t million people per decade, with t being measured in decades from 1810.If the population P(t) was 7.2 million people in 1810, estimate the population in 1820 (one decade later) by considering the work in example 2. A: P(1)= 8.375309 Q2. Given the CO2 addition rate shown in exampl.. 2022. 12. 6. [Notes & HW Answers] 5.2 The Definite Integral [Prepwork 5.2] Q1. Following example 1 in the text, suppose we considered n=4 to approximate the integral. What are each of the following? A: Δt= 0.25; f(t2)= 1/1.5 Q2. Find the exact value of the following integral. A: ∫^2_(-2) (4-x^2)^(1/2) dx = 2π Estimate the total area between f(x)=cos(x4) and the x-axis for (0≤x≤π)^(1/4)(Note: you will want to sketch the curve y=f(x). Shade the regions be.. 2022. 12. 6. [Notes & HW Answers] 5.1 How Do We Measure Distance Traveled? [Prepwork 5.1] Q1. Consider the velocity data for the car shown in table 5.1. Find lower and upper estimates for the distance it travels between 𝑡=4 and 𝑡=8. A: lower = 164 upper = 184 Write the difference between these as the following product: difference = (Δ𝑡)(Δ𝑣)= (2) (10) Q2. The velocity of a bicycle is given by v(t) = 4t feet per second, where t is the number of seconds after the bike sta.. 2022. 12. 6. [Notes & HW Answers] 4.6 Related Rates [Prepwork 4.6] Q1. In example 4, how fast is the fuel consumption changing if instead the car is moving 30 mph and decelerating at a rate of 3000 miles/hr^2? Answer: Decreasing at a rate of approximately (choose the closest answer) 600 mpg/hr. Q2. A spherical snowball is melting. Its radius decreases at a constant rate of 3 cm per minute from an initial value of 100 cm. How fast is the volume de.. 2022. 12. 6. [Notes & HW Answers] 4.5 Applications to Marginality [Prepwork 4.5] Q1. What is the relationship between (total) cost and marginal cost? A: Marginal cost is the derivative of total cost Q2. Suppose C and R are the cost and revenue functions for a particular product. Check all the statements below that are true. A. The critical points of the profit function happen when 𝑀𝑅=𝑀𝐶 or one of these derivatives does not exist. B. The profit is maximized whe.. 2022. 12. 6. [Notes & HW Answers] 4.4 Families of Functions and Modeling 2022. 12. 6. [Notes & HW Answers] 4.3 Optimization and Modeling [Prepwork 4.3] Q1. Consider the following problem: You are standing on a river bank and want to get to a house that is on the other side of the river and 800 feet up the river as quickly as possible. The river is 100 feet wide. You can walk along the river at a speed of 5 ft/sec or swim across the river at a speed of 3 ft/sec. What path should you take? A: Which of the following quantities are y.. 2022. 12. 6. [Notes & HW Answers] 4.2 Optimization [Prepwork 4.2] Q1. Which of the following examples satisfy the hypotheses of the Extreme Value Theorem on the given interval? (A) k(x)={3x^2+9 for 0≤xf(x)=2x^3−12x^2+18x+16. Find the minimal value of f(x) on the interval −1≤x≤2. Answer: The minimum value of f(x) on this interval is −16 and occurs at x=−1. Q3. Suppose f(x)f(x) is a continuous function defined on −∞𝑔(𝑡)=6𝑡𝑒^(−5𝑡) if 𝑡>0. A: global.. 2022. 12. 6. 이전 1 2 3 다음
