Question

To fill in the blanks, choose one of the most appropriate words from increasing, decreasing, concave up, concave down, positive and negative. a. If f′(x)>0 then the function f(x) is b. If f′(x)<0 then the function f(x) is c. If f′′(x)>0 then the function f(x) is d. If f′′(x)<0 then the function f(x) is e. If f(x) is concave up then f′′(x) must be f. If f(x) is decreasing then f′(x) must be 2. Data for a car is given below in a table where t represents time in minutes and d= f(t) represents distance traveled by the car. Find average rate of change of the distance over intervals [2,4],[4,6] and [2,6]. Which value is the best approximation of f′(4). Also. give a complete sentence interpretation of f′(4) in this context. 3. A capacitor is discharged through an electrical circuit. The amount of charge remaining in the capacitor after t seconds is eiven bv a=f(t) micro-Coulombs (uC). Estimate f′(2) and include units with your answer. Also, estimate the amount of charge remaining in the capacitor after 1.5sec. 4. The distance (in miters) traveled by an object is given by the formula, d=1.5(3)t where t is measured in seconds. You are also given that 1≤t≤3. Estimate to the nearest 0.01, the velocity of the object at t=2. Use the correct unit with your answer. 5. Use the limit definition of derivative to find f′(3) where f(t)=2t²−4t. 6. Daily demand of gasoline in a small village is D gallons at it sales for p dollars per gallon. Assume that D=f(p). Explain f′(p) in this context. Give units for f′(p). Consider the table Estimate and interpret f′(2.70)

Answer 1

In calculus, the signs of the derivative and second derivative of a function can tell us about the shape and behavior of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

Step-by-step explanation:

a. If f′(x)>0 then the function f(x) is increasing.

b. If f′(x)<0 then the function f(x) is decreasing.

c. If f′′(x)>0 then the function f(x) is concave up.

d. If f′′(x)<0 then the function f(x) is concave down.

e. If f(x) is concave up then f′′(x) must be positive.

f. If f(x) is decreasing then f′(x) must be negative.