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Differential and Integral Calculus Questions: 1. Use transformations to sketch the graphs of the following functions. Explain each step. y = ln (x − 2) + 3 y = − ln (−x) y = | ln (x)| y = ln |x| 2. If f(x) = 4x − 2 and g(x) = 1 x+2 , express the following as rational functions: fog(x) gof(x) gog(x) 3. Describe the motion of a particle that is at position (x(t), y(t)) at time t ≥ 1, if x(t) = ln t and y(t) = sin t. 4. Evaluate each of the following limits or explain why it does not exist: limx→4 x 2−16 x−4 limx→1 x 3−1 x−1 limt→0 sin(2t) t limx→3− x x−3 limx→∞ 3−2x x+1 limx→0 ln(1 + x) sin2 (π/x). 5. For what value(s) of the constant a is f(x) =  ax + 1 if x ≤ 1 (ax) 2 − 1 if x > 1 continuous on (−∞, ∞)? 6. Use the Intermediate Value Theorem to prove that there is a positive number c such that c 2 = 2. 7. For each of the following functions: f(x) = x + 3 x 2 − 2x − 15 g(x) =  x+3 x2−2x−15 if x 6= −3 −1 if x = −3 1 (a) Sketch its graph. (b) Find all values of x for which the function is NOT continuous. (c) Say why the definition of continuity is not satisfied for those values of x. 8. Let f(x) = x 2−3x+2 x2+3x−10 find: limx→0 f(x) limx→2 f(x) limx→−5 f(x) limx→∞ f(x). 9. Find the derivative dy dx for each of the following: y = cos 3x 2 y = x q x + 1 2x y = 2x x− √ x y = sec2 3x y = sin (sin (sin x)) y = x 2 sin 7x

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Differential And Integral Calculus
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