CalcExercisesCh3

CalcExercisesCh3 - (Exercises for Section 3.1: Derivatives,...

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(Exercises for Section 3.1: Derivatives, Tangent Lines, and Rates of Change) E.3.1 CHAPTER 3: DERIVATIVES SECTION 3.1: DERIVATIVES, TANGENT LINES, and RATES OF CHANGE In these Exercises, use the Limit Definition of the Derivative. Do not use the short cuts that will be introduced in later sections. 1) Let fx () = 5 x 2 + 1 . Find ± f 3 () . 2) Let = 3 x ± 2 . Consider the graph of y = in the usual xy -plane. a) Find the slope of the tangent line to the graph of f at the point a , fa , where a is an arbitrary real number in 2 3 , ± ² ³ ´ µ · . b) Find an equation of the tangent line to the graph of f at the point 9, f 9 . c) Find an equation of the normal line to the graph of f at the point 9, f 9 . 3) The position function s of a particle moving along a coordinate line is given by st = 2 t ± 3 t 2 , where time t is measured in seconds, and is measured in centimeters. a) Find the average velocity of the particle in the following time intervals: i. 1, 1.1 ± ² ³ ´ i i . 1, 1.01 ± ² ³ ´ b) Find the [instantaneous] velocity of the particle at time t = 1 .
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(Exercises for Section 3.2: Derivative Functions and Differentiability) E.3.2 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY 1) Let fx () = 1 x 2 . a) Use the Limit Definition of the Derivative to find ± . b) Use the shortcuts to find ± . 2) Let gw = 3 w 2 ± 5 w + 4 . a) Use the Limit Definition of the Derivative to find ± . b) Use the shortcuts to find ± . 3) Let rx = x 4 . a) Use the Limit Definition of the Derivative to find ± . Hint: Use the Binomial Theorem from Section 9.5 in the Precalculus notes. b) Use the shortcuts to find ± . 4) Let = 9 x 2 3 . Find ± , ±± , ±±± , and f 4 x . Do not leave negative exponents in your final answers. You do not have to simplify radicals or rationalize denominators. 5) Let y = x 10 . What is d 20 y dx 20 ? You shouldn’t have to show any work!
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(Exercises for Section 3.2: Derivative Functions and Differentiability) E.3.3 6) The position function s of a particle moving along a coordinate line is given by st () = 4 t 3 + 15 t 2 ± 18 t + 1 , where time t is measured in minutes, and is measured in feet. a) Determine the velocity function [rule] vt . (Use the short cuts.) b) Determine the velocity of the particle at times t = 1 , t = 2 , and t = 0 . c) Determine the acceleration function [rule] at . (Use the short cuts.) d) Determine the acceleration of the particle at times t = 1 , t = 2 , and t = 0 . 7) For each part below, answer “Yes” or “No.” a) If fx = x 4 ± 3 x + 1 , is f differentiable everywhere on ± ? b) If gx = 3 x ± 8 , is g differentiable at 8 3 ? c) If ht = t 2 3 , is h differentiable at 0? d) If px = 4 x + 3, if x ±² 1 x 2 ² 1, if x > ² 1 ³ ´ µ , is p differentiable at ± 1 ? e) If qx = 3 x + 2 , is q differentiable on the interval ± 10, 0 ? f) If = 3 x + 2 , is q differentiable on the interval 0,10 ? 8) Determine whether or not the graph of f has a vertical tangent line at the point 0, 0 and whether f has a corner , a cusp , or neither at 0, 0 .
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CalcExercisesCh3 - (Exercises for Section 3.1: Derivatives,...

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