Write an expression for the instantaneous velocity of the object at time t ! a.
Suppose an object moves along a straight line with positionį ! t " at time t. Write an expression for the slope of the tangent line to the curve y ! f ! x " at the point ! a, f ! a "".ġ66 |||| CHAPTER 2 LIMITS AND DERIVATIVESġ0. What does the Intermediate Value Theorem say?ĩ. (b) What does it mean for f to be continuous on the interval !$*, *"? What can you say about the graph of such a function?Ĩ. (a) What does it mean for f to be continuous at a? Which of the following curves have vertical asymptotes? Which have horizontal asymptotes?ħ. (b) What does it mean to say that the line y ! L is a horizontal asymptote of the curve y ! f ! x "? Draw curves to illustrate the various possibilities.Ħ. Makes with the positive direction of the x-axis. The angle of inclination of ! is the angle Let ! be the tangent line to the parabola y ! x 2 at the point (c) Sketch a graph of the derivative of T.ĥ7.
(b) Describe how the rate of change of T with respect to t varies as t increases. (a) Sketch a possible graph of T as a function of the time t that has elapsed since the faucet was turned on. When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. (b) The derivative of an odd function is an even function.ĥ6. (a) The derivative of an even function is an odd function. Recall that a function f is called even if f !$ x " ! f ! x " for all x in its domain and odd if f !$ x " ! $ f ! x " for all such x. (a) Find f ! $ !4 " and f ! # !4 " for the functionĥ5. (a) What does it mean to say that the line x ! a is a vertical asymptote of the curve y ! f ! x "? Draw curves to illustrate the various possibilities. Describe several ways in which a limit can fail to exist. Explain what each of the following means and illustrate with a sketch.Ģ. Then f !! a " exists if and only if these onesided derivatives exist and are equal. The left-hand and right-hand derivatives of f at a are defined by (b) For what values of x is f differentiable?ĥ4. Where is the greatest integer function f ! x " ! ) x * not differentiable? Find a formula for f !and sketch its graph.ĥ3 (a) Sketch the graph of the function f ! x " ! x ( x (. Find a formula for f !and sketch its graph.ĥ2. Show that the function f ! x " ! ( x $ 6 ( is not differentiable at 6. (d) Illustrate part (c) by graphing y ! x 2 ' 3.ĥ1. (c) Show that y ! x 2 ' 3 has a vertical tangent line at !0, 0 ". (a) If t! x " ! x 2 ' 3, show that t!!0 " does not exist. (c) Show that y ! s 3 x has a vertical tangent line at !0, 0 ". (a) If a # 0, use Equation 2.7.5 to find f !! a ". A Numbers, Inequalities, and Absolute Values.11.11 Applications of Taylor Polynomials.
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