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# math 7 Math 191G

NMSU

GPA 3.9

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This 4 page Class Notes was uploaded by Brenna Graham on Sunday October 9, 2016. The Class Notes belongs to Math 191G at New Mexico State University taught by Louiza Fouli in Fall 2016. Since its upload, it has received 3 views. For similar materials see Calculus I in Math at New Mexico State University.

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Date Created: 10/09/16

Exam 1 Review Fall 2016. This does not need to be handed in. It is to assist you in studying. 1. Let s(t) = t + 2t + 3 be the position function for an object moving along a line. (a) Find the average velocity of the object over the interval [1;3] (b) Find the average velocity of the object over the interval [1;t]: Simplify your answer. (c) Estimate the instantaneous velocity of the object at time t = 1: 2. Evaluate the following limits. Justify your anwers. 2 (a) limx + x ▯ 6 x!2 x ▯ 2 x + 5x + 4 (b) lim 2 x!▯4 x + 3x ▯ 4 p (c) lim x + 2 ▯ 3 x!7 x ▯ 7 2 2 h + 3▯ 3 (d) lim h!0 h 1 1 (e) lim( ▯ ) x!0 t t + t 3 2x ▯ 3x (f)x!1m 4x + 5x + 1 x (g)x!▯1m p 2 x + 1 tanx (h) lim x!0 tan2x x (i) lim x!0 cosx ▯ 1 sinx + 2x (j) lim x!1 x + 1 3 3 3 (k) lim lim lim x!0+ ex▯ 1 x!0▯ ex▯ 1 x!0 ex▯ 1 x ▯ 4 x ▯ 4 x ▯ 4 (l)x!0+ x2 x!0▯ x 2 x!0 x2 2 1 (m) lim x cos( 2) x!0 x 1 8 p 2 < x + 1 if x < 0 3. Let f(x) = 3 ▯ x if 0 ▯ x < 3 : 2 (x ▯ 3) if x > 3 (a) Evaluate each limit, if it exists. i. lim+f(x) x!0 ii.lim f(x) x!0 ▯ iii. lim f(x) x!0 iv. lim f(x) x!3 ▯ v. lim+ f(x) x!3 vi. lim f(x) x!3 (b) Determine the discontinuities of f . (Justify your answer using the de▯niton of continuity.) (c) Determine if f is right or left continuous at each of the discontinuity. (Justify your answer using the de▯nitons of right or left continuity.) 4. The graph of f(x) is shown below. Use the graph to determine the following: (a) lim f(x) x!▯1 ▯ (b)x!▯1m+f(x) (c) lim f(x) x!▯1 (d) f(▯1) (e) f(3) (f) lim f(x) x!3 (g) the discontinuities of f(x) (Justify your answer using the de▯nition of continuity.) 3 5. Show that the equation x + 3x ▯ 1 = 0 has a solution. Justify your reasoning. 2 6. The graph of y = f(x) is shown below. Find the following. (a) f (▯2) (b) f (0) (c) the equation of the tangent line to y = f(x) at x = 0 (d) f (2) (e) the equation of the tangent line to y = f(x) at x = 2 (f) estimate f (3) 7. Use the limit de▯nition of the derivative to ▯nd f (x) if (a) f(x) = 3x ▯ 2x + 5 4 (b) f(x) = p (c) f(x) = 2x + 1 8. Use the derivative rules to compute the derivative of the given function. 5 3 2 3 (a) f(x) = ▯ 4 x + p x x x 2 (b) f(x) = e (x + 4x) ▯ ▯1 ▯ p (c) f(x) = x + 5 ( x ▯ 3) (d) f(x) = 2x + 5 x ▯ 3x + 6 2 x + sinx (e) f(x) = 2 x + cosx 3 2 x (f) f(x) = x sinx(1 + e ) (g) f(x) = xlnx sinx x 5e ▯ 1 (h) f(x) = x + 1 9. Find the equation of the tangent line to the given function at the given point. (a) the function in 8a. at x = 1 (b) the function in 8e. at x = 0: 10. Find where the function has a horizontal tangent line (i.e. f (x) = 0) 2 x (a) f(x) = x e 5 3 (b) f(x) = x ▯ x + 6 1 1 (c) f(x) = x + x ▯ x + 3 3 2 4

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