solve basic limits, derivative, and continuity

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MATH 211 Learning Activity – The -δ Definition of the Limit Name:   4 3 − x = −5. x→10 5 1. Prove that lim Fall 2017 MATH 211 Learning Activity – The -δ Definition of the Limit  2. Prove that lim x2 + 2x − 7 = 1. x→2 Fall 2017 MATH 211 3. Prove that Learning Activity – The -δ Definition of the Limit 5 = −∞. x→−1− (x + 1)3 lim Fall 2017 MATH 211 Learning Activity – Continuity Fall 2017 Name: 1. Determine if the following function is continuous at a = 4. (If it is discontinuous state which condition for continuity has not been met.) f (x) = 2. Consider the function h(t) = x2 5x − 2 − 9x + 20 t2 + 3t − 10 . Redefine h(t) as a piecewise function that is continuous. t−2 MATH 211 Learning Activity – Continuity Fall 2017 1 3. Consider the functions f (x) = x2 − 1 and g(x) = . Determine whether or not each of the following x composite functions are continuous at 0. (Explain your answers.) (a) f ◦ g (b) g ◦ f (c) f ◦ f (d) g ◦ g  2   x +x 4. Let g(x) = a   4 − x2 if if if x1 (a) Determine the value of a for which g is continuous from the left at 1. (b) Determine the value of a for which g is continuous from the right at 1. (c) Is there a value of a for which g is continuous at 1? Justify your answer. MATH 211 Learning Activity – Continuity Fall 2017 5. Where is the following function continuous? (Hint: There should be 3 points of discontinuity.) y = csc(2x) on the interval [0, π] MATH 211 Learning Activity – Introducing the Derivative Name: 1. Let f (x) = 3x2 + 2x − 10. (a) Find f 0 (1). (b) Find the equation of the line tangent to f (x) at x = 1. 2. Let f (x) = 1 . Find the slope of the line tangent to f (x) at the point (−1, −1). x Fall 2017 MATH 211 Learning Activity – Introducing the Derivative Fall 2017 3. If a rock is thrown upward on the planet Mars with a velocity of 10m/s, its height (in meters) after t seconds is given by the function s(t) = 10t − 1.8t2 . Answer the following: (a) Find the velocity of the rock after one second. (b) Find the velocity of the rock when t = a. (c) When will the rock hit the surface? (d) With what velocity will the rock hit the surface? MATH 211 Learning Activity – The Derivative as a Function Name: 1. Find f 0 (x) if f (x) = x3 − x. Fall 2017 MATH 211 Learning Activity – The Derivative as a Function Fall 2017 2. The graphs of four derivatives are given below. Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV. (a) (c) (b) (d) MATH 211 Learning Activity – The Derivative as a Function 3. For the function graphed below: (a) At what points does the derivative not exist and why? (b) Identify the points at which the function is continuous but not differentiable. (c) Identify the intervals over which the function is differentiable. Fall 2017 MATH 211 Learning Activity – Derivative Rules Fall 2017 Name: 1. Find the derivatives of the functions. (Simplify by combining like terms and leaving no negative exponents.) (a) f (t) = t100 (b) f (x) = 1 x4 (c) g(u) = u2/3 (d) y = 3x5 + 4x3 − 12x + 33 2. Find f 0 (x), f 00 (x) and f (3) (x) for f (x) = 18 x4 − −3x2 + 1. MATH 211 Learning Activity – Derivative Rules 3. Let f (x) = x2 − 6x + 5 (a) Find the values of x for which the slope of the curve y = f (x) is 0. (b) Find the values of x for which the slope of the curve y = f (x) is 2. 4. Assume g 0 (4) = 1. Evaluate d [2x − 3g(x)] dx x=4 Fall 2017
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1_7: Defination of limit
1.

4
5

lim (3 − 𝑥) = −5

𝑥→10

Solution
Replace the value of x with 10
4
(3 − (10)) = −5
5
2. Prove that lim (𝑥 2 + 2𝑥 − 7) = 1
𝑥→2

Solution
Replace x with 2
22 + 2(2) ...


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