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[Umich] COE Core/MATH 115 (Calc 1)26

[Notes & HW Answers] 4.1 Using First and Second Derivatives [Prepwork 4.1] Q1. Suppose f(x)f(x) is a continuous function that has a local maximum at the point (3,2).Which of the following statements must be true? A: (1) If a is just a little greater than 3, then f(a) ≤ 2. (2) f(x) has a critical point at x=3. (3) If a is just a little less than 3, then f(a) ≤ 2. Q2. Suppose f(x)f(x) is a continuous function that has a local minimum at (1,6)(1,6) and no o.. 2022. 12. 6.
[Notes] 3.10 Theorems About Differentiable Functions 2022. 11. 29.
[Notes & HW Answers] 3.9 Linear Approximation [Prepwork 3.9] Q1. What is the value at x=.1 of the local linearization L(x) of (1+x)^5 at x=0? A: L(0.1) = 1.5 Q2. Suppose you use the tangent line approximation near t=4 to estimate s′(4.05), where s(t) is a function that is always concave up. Then your answer is an A: underestimate Q3. Suppose the linear approximation of g(x)g(x) near x=6 is given by L(x)=5(x−6)−12. A:Then g(6) = -12 and g'(6.. 2022. 11. 21.
[Notes & HW Answers] 3.7 Implicit Functions [Prepwork 3.7] Q1. Which of the following points satisfies the equation 𝑥^2−𝑥𝑦^2=2? A: (1) (-2^(1/2)/2, (4.5)^(1/4)) ; (2) (-2^(1/2),0) Q2. (1) Find dy/dx when x^2 - xy^2 =2. A: (1) dy/dx = (y^2-2x)/-2xy (2) Find an equation for the tangent line to x^2 - xy^2 = 2 at the point (2,1). A: (2) y = (3/4)(x-2) + 1 Q3. Find all points satisfying the equation 𝑥2−𝑥𝑦2=2x2−xy2=2 where the tangent line is v.. 2022. 11. 21.
[Notes & HW Answers] 3.6 The Chain Rule and Inverse Functions [Prepwork 3.6] Q1. Let 𝑔(𝑥)g(x) be an invertible, differentiable function with values given in the table below. 𝑥 2 5 8 𝑔(𝑥) 8 2 0 𝑔′(𝑥) -2 -0.25 -0.1 Find a formula for the tangent line of 𝑔^(−1)(𝑥) at 𝑥=2. A: y = -4(x-2)+5 Q2. Find the derivative of 𝑓(𝑥)=arctan(5ln(𝑥)). A: f'(x) = 5/(x*(25*ln^2(x)+1) [HW 3.6] Q1. Find the derivative of the function f(t), below. f(t) = ln(t^9 +8) A: f'(t) = 9t^.. 2022. 11. 21.
[Notes & HW Answers] 3.5 The Trigonometric Functions [Prepwork 3.5] Q1. The Bay of Monterey in California is known for extreme tides. The depth of the water, y, in meters can be modeled as a function of time, t, in half-hours after midnight, by y=12+6cos(t). How quickly is the depth of the water rising or falling at 3 a.m.? (Make sure you compute this in radians and give your answers to three decimal places.) Rising at (in m/half-hours) A: 1.676 Q.. 2022. 11. 21.
[Notes & HW Answers] 3.4 The Chain Rule [Prepwork 3.4] Q1. The length,L, in micrometers (𝜇μm), of steel depends on the air temperature degrees celsius, and the temperature depends on time,t, measured in hours. If the length of a steel bridge increases by 0.25μm for every degree increase in temperature, and the temperature is increasing at 4 degrees celsius per hour, how fast is the length of the bridge increasing? A: At a rate of 1 μm.. 2022. 11. 21.
[Notes & HW Answers] 3.3 The Product and Quotient Rules [Prepwork 3.3] Q1. Which of the following are possible formulas for a function y = f (x) such that f'(x) = 10x^9*e^x+x^10*e^x? • A. x10ex+10 • B. 3+x10ex • C. x10ex+1 • D. 9x9ex−1 • E. x10ex+10 • F. 10x10ex • G. x10ex A: ABG Q2. Which (single) rule would one use to differentiate f (x) =x*e^x? A: the Product rule [HW 3.3] Q1. Find the derivative of the function f (x), below. It may be to your adv.. 2022. 10. 30.
[Notes & HW Answers] 3.2 The Exponential Functions [Prepwork 3.2] Q1. Let f (x) = x^10 +10^x. Find the first and second derivatives of f(x). A: f'(x) = 10x^9 + (ln 10)10^x ; f''(x) = 90x^8+(ln 10)(ln10)10^x Q2. For what value of a is y = 1+5x the tangent line for y = ax at x = 0? A: a = e^5 [HW 3.2] Q1. Find the derivative of y = 4x^8+8^x+12. A: dy/dx = 32x^7 + (ln 8)8^x Q2. Find the derivative of z = (ln 10)e^x. A: dz/dx = (ln 10)e^x. Q3. Find .. 2022. 10. 28.
[Notes & HW Answers] 3.1 Powers and Polynomials [Prepwork 3.1] Q1. For the derivative of 6x^6+4x^5+1x^4, what is the value of the coefficient of x^5? A: the coefficient is 36. Q2. Find the slope of the tangent line to f (x) = x^3 +x^2 +6/x at the point where x = 1. A: -1. [HW 3.1] Q1. Find the derivative of h(q) = 1/(q)^(1/4) . A: h'(q) = (-1/4)*(1/(q)^(4/5) Q2. Find the derivative of y = 10t^5−8t^(1/2) + 9/t. A: dy/dt = 50t^4-4t^(-1/2)-9t^(-.. 2022. 10. 28.
[WeBWork] Prepwork & HW 2.1~2.6 & 3.10 Answers Keywords: derivative, speed, acceleration, instantaneous velocity, changing, derivative function, the second derivative, differentiability Questions Example: [Prepwork 2.4] Q2. The cost of extracting T tons of ore from a copper mine is C = f (T) dollars. What does it mean to say that f 0(3400) = 250? A. When 3400 tons of ore have already been extracted from the mine, the cost of extracting the n.. 2022. 10. 28.
[Notes] Ch.2 Key Concept: The Derivative Keywords: derivative, speed, acceleration, instantaneous velocity, changing, derivative function, the second derivative, differentiability Preview the notes: Keywords: derivative, speed, acceleration, instantaneous velocity, changing, derivative function, the second derivative, differentiability 2022. 10. 28.

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