1.1 - Can Change Occur at an Instant?

A particle is moving on the x-axis and the position of the particle at time t is given by x(t), whose graph is shown above. Which of the following is the best estimate for the speed of the particle at time t=4?

5 Since the position function is linear, the slope of the graph at time t in the interval [0,6] is the speed of the particle at time t=4.

A particle is moving on the x-axis, and the position of the particle at time t is given by x(t), whose graph is shown above. Which of the following is the best estimate for the speed of the particle at time t=6?

5/4 Since the position function is linear, the slope of the graph at time t in the interval [0,8] is the speed of the particle at time t=6.

A car is driven on a straight road, and the distance traveled by the car after time t=0 is given by a quadratic function s, where s(t) is measured in feet and t is measured in seconds for 0≤t≤12. Of the following, which gives the best estimate of the velocity of the car, in feet per second, at time t=6 seconds?

s(7)−s(5)7−5 This gives the average velocity of the car over the shortest given time interval containing t=6.

An automobile is driven on a straight road, and the distance traveled by the automobile after time t=0 is given by a quadratic function s, where s(t) is measured in feet and t is measured in seconds for 0≤t≤12. Of the following, which gives the best estimate of the velocity of the automobile, in feet per second, at time t=8 seconds?

s(9)−s(7)9−7 This gives the average velocity of the automobile over the shortest given time interval containing t=8.

The graphs of the functions f and g are shown above on the interval 0≤x≤5. (a) Write a difference quotient that best approximates the instantaneous rate of change of g at x=2.5. (b) Let h be the function defined by h(x)=g(f(x)). Find the value of limx→3h(x) or explain why the limit does not exist. Use correct limit notation in your answer. (c) Let k be the function defined by k(x)=(4−f(x))g(x). Consider x=2 and x=4. Determine whether k is continuous at each of these values. Justify your answers using correct limit notation. (d) Values of f(x) at selected values of x are given in the table below. What type of discontinuity does f have at x=1 ? Based on the values in the table, what is a reasonable estimate for limx→1f(x) ? Give a reason for your answer.

A - g(3)−g(2)3−2=1−43−2 B - As x approaches 3 from the left and from the right, f(x) approaches 2 from above. Therefore, the limit of g(x) should be evaluated as x approaches 2 from the right, as shown. limx→3−h (x)=limx→3−g (f (x))=limx→2+g (x)=4 limx→3+h (x)=limx→3+g (f (x))=limx→2+g (x)=4 Therefore, limx→3h (x)=4. C -

The graphs of the functions f and g are shown above on the interval −3≤x≤5. The graphs consist of line segments, except where f(x)=5sin(πx)2−2x3 on the interval 0≤x≤2. (a) Write a difference quotient that best approximates the instantaneous rate of change of g at x=4.5. (b) Let h(x)=g(f(x)). Find limx→4h(x). Use correct limit notation in your answer. (c) Let k(x)=f(x)+g(x). Consider x=−1, x=2, and x=4. At which one of these values does k have a jump discontinuity? Justify your answer for the jump discontinuity using correct limit notation. (d) Values of f(x) at selected values of x are given in the table below. What type of discontinuity does f have at x=1 ? Based on the values in the table, what is a reasonable estimate for limx→1f(x) ? Give a reason for your answer.

A - g(5)−g(4)5−4=(−1)−(−3)5−4 B - As x approaches 4 from the left and from the right, f(x) approaches 2 from below. Therefore, the limit of g(x) should be evaluated as x approaches 2 from the left, as shown. limx→4−h (x)=limx→4−g (f (x))=limx→2−g (x)=1 limx→4+h (x)=limx→4+g (f (x))=limx→2−g (x)=1 Therefore, limx→4h (x)=1. C - There is a jump discontinuity at x=−1 because both one-sided limits exist at x=−1 and are finite but are not equal: limx→−1−k (x)=limx→−1−f (x)+limx→−1−g (x)=0+(−1)=−1 limx→−1+k (x)=limx→−1+f (x)+limx→−1+g (x)=1+1=2 Note: k does not have a jump discontinuity at x=2 limx→2−k (x)=0+1=1 and limx→2+k (x)=−2+3=1 Note: k does not have a jump discontinuity at x=4. limx→4−k (x)=2+(−3)=−1 and limx→4+k (x)=2+(−3)=−1 D - The graph of f indicates that f has a removable discontinuity at x=1. The graph shows that f is decreasing near x=1, so 2.621>limx→1f (x)>2.615. Since the table of values reflects approximate linear behavior, a reasonable estimate for limx→1f (x) x would be determined by averaging the function values on each side of 1. The estimate is as follows. limx→1f (x)≈f (0.999)+f (1.001)2=2.621+2.6152=2.618 Alternately, a reasonable estimate can be found by observing that as x increases by 0.001 f (x) decreases by 0.002 or 0.003. Since 2.618 is 0.003 from both 2.621 and 2.615, it is the best estimate.

The size of a population of fish in a pond is modeled by the function P, where P(t) gives the number of fish and t gives the number of years after the first year of introduction of the fish to the pond for 0≤t≤10. The graph of the function P and the line tangent to P at t=4 are shown above. Which of the following gives the best estimate for the instantaneous rate of change of P at t=4 ?

The slope of the line joining (3.9,P(3.9)) and (4.1,P(4.1)) If h>0 is close to 0, we expect the slope of the line joining (4−h,P(4−h)) and (4+h,P(4+h)) to be close to the slope of the tangent line at t=4.

The size of a population of bacteria is modeled by the function P, where P(t) gives the number of bacteria and t gives the number of hours after midnight for 0≤t≤10. The graph of the function P and the line tangent to P at t=8 are shown above. Which of the following gives the best estimate for the instantaneous rate of change of P at t=8 ?

The slope of the line joining (7.9,P(7.9)) and (8.1,P(8.1)) If h>0 is close to 0, we expect the slope of the line joining (8−h,P(8−h)) and (8+h,P(8+h)) to be close to the slope of the tangent line at t=8.