MIDTERM EXAM SOLUTIONS

Transcription

1 MIDTERM EXAM SOLUTIONS Directions and rules. The eam will last 70 minutes; the last five minutes of class will be used for collecting the eams. No electronic devices of any kind will be allowed, with one eception: a music player that nobody else can hear, and whose controls you do not use during the eam (just put it on shuffle). Please turn your cell phone off. You are not allowed to use books or notes. There are ten questions on this eam, each worth 10 points. Correct answer earns full credit. Wrong answers might earn partial credit if some steps are correct. 1. A small rocket is fired vertically upwards, starting at ground level. The rocket engine fires for a few seconds and then the rocket falls back to the ground. While the rocket engine is firing, its upwards velocity increases linearly. (a) After the rocket engine stops firing, does the rocket s velocity increase or decrease? It decreases, because gravity slows it down. (b) When the rocket reaches its highest point, what is its velocity? Zero. (c ) When the rocket is falling, is its velocity increasing or decreasing? Decreasing. It is getting more negative. (d) When the rocket is falling, is its speed increasing or decreasing? Increasing. It is falling faster and faster, like every object that falls under the influence of gravity. (e) Sketch a graph showing its velocity as a function of time. (f) Sketch a graph showing its height as a function of time. Please draw this graph directly BELOW the velocity graph so corresponding times line up vertically. Hint: The maimum height does not occur at the same moment the engine stops firing. Answer. The graphs are on the net page (They won t fit on the first page). Also the velocity graph is printed below the position graph instead of the other way around as was requested. The velocity has to be linear on each interval, with positive slope while the rocket is firing and negative slope when the rocket is just falling. That is, at first it goes faster and faster upwards, then, the velocity is still positive but decreasing; when the velocity reaches zero the rocket is at its highest point; then the velocity goes negative as the rocket falls. The height graph is composed of pieces of two parabolas, the first one concave up and the second one concave down. It should end up with height zero, when the rocket hits the ground. 1

2 2 MIDTERM EXAM SOLUTIONS

3 MIDTERM EXAM SOLUTIONS 3 2. Above is the graph of a function f. What is the value (if any) of (a) 1 + f() Answer: 2 (b) 1 f()answer: 1 (c) 1 f() Answer: undefined, or does not eist (d) 0 + f()answer: 3. Consider the function f() = Are there any values of at which f is not continuous? If so, which values? Answer: At the zeroes of the denominator, namely = ±2.

4 4 MIDTERM EXAM SOLUTIONS In problems 4 through 8, evaluate the it, and show your work (unless, of course, only one step is needed). ( ) ( 1)( + 1) = = = 2 2 (4 2 ) = 2 0 = = We can drop the term in the numerator since it s the leading terms that determine the it; here that is the lower-degree terms: 1 dominates for small. If instead you just break up the fraction you get 0 ( ) = + undefined, which would give you undefined, but is the (more) correct answer. 7. tan 0 sin 0 cos sin 0 cos sin 0 cos 0 = 1 1 = 1

5 MIDTERM EXAM SOLUTIONS = 2 3 Since numerator and denominator have the same degree, the ratio of the leading coefficients is the it. Alternately you could divide numerator and denominator by 5 and use the it rules ( )( ) ( ) 1 + (1 ) 0 ( ) 0 = = 0 2 = We learned that the function sin has it 1 as goes to zero. In order to show that directly from the definition of it, the inequalities < δ and sin 1 < ɛ will be involved. How eactly will the argument go? (Pick one of the choices below.) (a) Suppose < δ and calculate ɛ in terms of δ to make sin (b) Suppose sin 1 < ɛ and calculate δ in terms of ɛ to make < δ. (c) Show how to calculate δ in terms of ɛ such that if < δ then sin (d) Show how to calculate ɛ in terms of δ such that if < δ then sin Answer: Choice c is correct.