[Three Differentiation Questions] Differentiation: Definition and Fundamental Properties Exam

References:

Based on your knowledge in these:

02.01 Average and Instantaneous Rates of Change and the Derivative Definition Segment: 1

02.02 Determining Differentiability and Estimating Derivatives Segment: 1

02.03 Derivative Rules: Constant, Sum, Difference, Constant Multiple, and Power Segment: 1

02.05 The Product Rule and the Quotient Rule Segment: 1

02.06 Derivatives of Trigonometric Functions Segment: 1

02.07 Derivatives of Exponential and Logarithmic Functions Segment: 1

Question:

Question 1 (Essay Worth 30 points) (0201 H0) The total amount of rainfall, in centimeters, as a function of time, in days, is modeled by the function f, where g is a differentiable function. The data in the table gives values of g(t) at selected values of t. J2 for 25MB 1 ' for 851514] 1(1) = l git) gm 1.875 1.433 1.115 1.004 Part A: What is the average rate of change of the amount of rainfall from day 8 to day 14, according to function f? (5 points) Part B: Use the data in the table to approximate f'(13). Show your work. (5 points) Part C: Is fa continuous function from 2 5 ts 14? Explain. (10 points) Part D: Find f'(5) using the limit denition of the derivative. Show your work and explain its meaning. (10 points) 1:: Question 2 (Essay Worth 30 points) (02.02 HC) When food is left out at room temperature for a long period of time, mold begins to grow on it. For 0 5 ts 40, the amount of mold on a cherry pie is modeled by the twice-differentiable function C, where C(t) is measured in millimeters and tis measured in days. Values of C(t) at selected values of time tare shown in the table. ((days) 0 1o 20 4o C(t) (millimeters) 0.357 1.323 4.906 67.431 Part A: Use the data in the table to approximate C(15). Show the computation that led to your answer. (10 points) Part 3: Using correct units, interpret the meaning of C(15) in the context of the problem. (10 points) Part C: The amount of mold is also modeled by the Mice-differentiable function D for 0 5 ts 40, where D(t) is measured in millimeters and time tis measured in days. It is known that D(t) can be modeled by the function D(t) = 0.357(1.14)', where D(t) is measured in millimeters and tis measured in days. Using graphing technology, nd the value of D'(15). (10 points) Choose File no file selected Question 3 (Essay Worth 40 points) (02.07 HC) The graph of continuous function ffor -3 S x S 16 is shown. 3 N -1 -2 -3 -4 -4 -2 2 6 8 10 12 14 16 18 20 Let g and h be functions defined by g(x) = 2* . f(x) and h(x) =3x2 +4x -InVx. Part A: Find g'(8). (10 points) Part B: The function m is defined by m(x) = f(x) . h(x). Find m'(14). (15 points) Part C: The function n is defined by n(x) = -2. Find n'(1). (15 points) f ( x )