Mix Questions - Worksheet 1

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Chapter 5 Class 12 - Continuity & Differentiability - Mix Questions Worksheet 1 by teachoo Chapter: Chapter 5 Class 12 - Continuity & Differentiability Name: _____________________________ School: _____________________________ Roll Number: _____________________________ 2-Mark Questions A digital thermostat controls an air conditioner. It turns the AC on when the temperature is 24^∘ C and off when it reaches 22^∘ C. Sketch a rough graph of the temperature over time and explain why the function describing the AC's power consumption is discontinuous. Explain in one sentence why a function with a "hole" (a removable discontinuity) is not continuous at that point, even if the limit exists. The function f(x)=|x| is continuous everywhere but not differentiable at x=0. What physical property of a path would this model? If the velocity of a particle is given by v(t)=2t, its position is s(t)=t^2. If the velocity is given by v(t)=|t|, is the position function s(t) still differentiable everywhere? Why or why not? Give a real-world example of two functions, f and g, where both are continuous, but their quotient f/g is not. The derivative of f(x)=sin⁡(x) is f^' (x)=cos⁡(x). Geometrically, what does it mean that when sin⁡(x) is at its maximum value, its derivative is zero? A company finds that its cost function C(x) is differentiable. Why is it useful for them to know that C^' (1000)=50 ? Is the function f(x)=x|x| differentiable at x=0 ? Justify your answer without calculation, by considering the "smoothness" of the graph. 3-Mark Question 9. The population of a bacterial colony is modeled by P(t)=100e^kt. Show that the rate of growth of the population is directly proportional to the population size at any given time t. What does this tell you about the nature of this type of growth? Important links Answer of this worksheet - https://www.teachoo.com/25574/5353/Mix-Questions---Worksheet-1/category/Teachoo-Questions---Mix/ Full Chapter with Explanation, Activity, Worksheets and more – https://www.teachoo.com/subjects/cbse-maths/class-12th/ Maths Class 12 - https://www.teachoo.com/subjects/cbse-maths/class-12th/ For more worksheets, ad-free videos and Sample Papers – subscribe to Teachoo Black here - https://www.teachoo.com/black/   Answer Key to Mix Questions Worksheet 1 Graph: The graph of temperature would oscillate between 22^∘ C and 24^∘ C. Explanation: The power consumption function is discontinuous because it jumps instantaneously from O (when the AC is off) to a positive value (when the AC is on), and vice-versa. There is no intermediate state. A function is continuous at a point only if the limit equals the function's value at that point; a "hole" means the function is undefined there, so this condition fails. This would model a path with a sharp, sudden change in direction, like turning a corner abruptly without slowing down. No. The position function s(t) would have a "corner" at t=0, making it not differentiable there. The velocity changes direction instantaneously from negative to positive. Let f(t) be the altitude of a rocket (a continuous function) and g(t) be the amount of fuel remaining (also continuous). The function f(t)/g(t) (altitude per unit of fuel) would be discontinuous when the fuel runs out, i.e., when g(t)=0. When sin⁡(x) is at its maximum value (a peak of the wave), the tangent line to the graph is perfectly horizontal, and a horizontal line has a slope of zero. This value, the marginal cost, tells the company that after producing 1000 units, the approximate cost to produce the next single unit is $50. Yes, it is differentiable. The function can be written as f(x)={■(x^2&" if " x≥0@-x^2&" if " x<0)┤. The graph is smooth and does not have a sharp corner at x=0. P^' (t)=d/dt (100e^kt )=100⋅e^kt⋅k=k⋅(100e^kt )=k⋅P(t). This shows the rate of growth P^' (t) is proportional to the population size P(t). This is the defining characteristic of exponential growth.