Pre-Calculus Section 12.4: Tangent Lines and Derivatives 1. Determine the interval on which the function in the graph below is decreasing.

Transcription

1 Pre-Calculus Section 12.4: Tangent Lines and Derivatives 1. Determine the interval on which the function in the graph below is decreasing. Determine the average rate of change for the function between the indicated values of the variable. 2. The graph of a function is sketched below. 4. The graph of a function is given below. What is the average rate of change of the function between the indicated values of the variable? Determine the interval on which the function is decreasing. a. b. [1, 3] c. [ 1, 1] d. e. [ 3, 1] 5. The graph of a function is sketched as follows: 3. The graph of a function is given as follows:

2 a. 0 b. 4 c. 2 d. 6 e What is the average rate of change of the function f (x) = x + x 2 between x = 0 and x = 5? a. 6 b. 7 c. 10 d. 8 e What is the average rate of change of the function f (x) = 3x 7 between x = 2 and x = 3? 8. A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. If P is the point (15, 282 ) on the graph of V, find the slope of the secant line PQ when Q is the point on the graph with t = 25. Explain the meaning of your answer. t (min) V (gal) A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. If P is the point (15, 279 ) on the graph of V, estimate the slope of the tangent line at P by averaging the slopes of two secant lines, passing through P and the points on the graph with t = 10 and t = 20. Explain the meaning of your answer. t (min) V (gal) If a ball is thrown into the air with a velocity of 35 ft/s, its height in feet after t. Find the average velocity for the time period beginning when t = 5 and lasting 0.03 s. Round to the nearest thousandth if necessary. a b c d e If an arrow is shot upward on the moon with a velocity of 45 m/s, its height in meters after t average velocity over the interval [1, 1.04].. Find the

3 OR, 12. The displacement (in feet) of a certain particle moving in a straight line is given by where = slope at a given x-coordinate where t is measured in seconds. Find the average velocity over the interval [1, 1.7]. 13. Find the slope of the tangent line to the graph of f at the point ( 5, 42 ). f ( x ) = 6x The Tangent line to the curve y = f(x) at the point is the line through P with slope 16. If a ball is thrown into the air with a velocity of 35 ft/s, its height in feet after t. Find the instantaneous velocity when t = If an arrow is shot upward on the moon with a velocity of 55 m/s, its height in meters after t instantaneous velocity after one second.. Find the slope at x = a would be The Average 18. Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. Instantaneous (EXACT) would be, a. at (5, 30) where = slope at x = a b. 15. The Tangent line to the curve y = f(x) at the point is the line through P with slope, where = slope at x = a

4 c. f ( x ) = 7 3x + x 2 a. f ' ( 5 ) = 15 b. f ' ( 5 ) = 14 c. f ' ( 5 ) = 13 d. f ' ( 5 ) = Find the slope of the tangent line to the graph of f at the point ( 1, 9 ). f ( x ) = 6 3x 25. Find the derivative and the equation of the tangent line at x = 3. g ( x ) = 4x 2 + x Find the slope of the tangent line to the graph of f at the point ( 1, 0 ). f ( x ) = 3 + 5x 8x Find the equation of the tangent line to the graph of f at the point whose x - coordinate is 1. f ( x ) = x 3 8x Find the slope of the tangent line to the graph of f at the point ( 3, 135 ). f ( x ) = 5x If a ball is thrown into the air with a velocity of 60 ft/s, its height (in feet) after t. Find the velocity when t = 5. a. 124 ft/s b. 115 ft/s c. 120 ft/s d. 117 ft/s e. 121 ft/s 22. Find an equation of the tangent line to the curve at x = The displacement (in meters) of a particle moving in a straight line is given by the equation of motion, which is measured in seconds. Find the velocity of the particle at t = Find the tangent line to y(x) at x = 1 and graph the curve and the tangent line on a graphing calculator. y = 4x x Find the derivative of the following function at If an arrow is shot upward on the moon with a velocity of 51 m/s its height (in meters) after t. When will the arrow hit the moon? Round the result to the nearest thousandth if necessary.

5 30. If an arrow is shot upwards on the moon with a velocity of 52 m/s its height (in meters) after t. When will the arrow hit the moon? Round the result to the nearest thousandth if necessary. 31. If an arrow is shot upward on the moon with a velocity of 61 m/s its height (in meters) after t. With what velocity will the arrow hit the moon? a. 58 b. 61 c. 62 d e If an arrow is shot upward on the moon, with a velocity of 54 m/s its height (in meters) after t what velocity will the arrow hit the moon?. With 33. A spherical balloon is being inflated. Find the rate of change of the surface area with respect to the radius r when r = 3 ft. Round your answer to nearest hundredth. 34. A spherical balloon is being inflated. Find the rate of change of the surface area with respect to the radius r when r = 2 ft. Round your answer to three significant figures. 35. A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. Find the average heart rate (slope of the secant line) over the time interval [ 40, 42 ]. t (min) Heartbeats 2,536 2,664 2,808 2,940 3, The cost (in dollars) of producing x units of a certain commodity is Find the instantaneous rate of change of with respect to x when x = 109. (This is called the marginal cost.) Explain the meaning of your answer. 38. If an arrow is shot upward on the moon with a velocity of 48 m/s, its height (in meters) after t. 37. What is the difference between a tangent line and a secant line? (a) Find the velocity of the arrow after one second. m/s (b) Find the velocity of the arrow when.

6 m/s (c) At what time t will the arrow hit the moon? BONUS: Show, with supporting mathematical evidence that this problem is either realistic or unreasonable in reality s (d) With what velocity will the arrow hit the moon? m/s 39. A roast turkey is taken from an oven when its temperature has reached 185 F and is placed on a table in a room where the temperature is 70 F. The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour. Please round the answer to the nearest tenth. 40. A curve has the equation y = g(x). Choose an expression for the slope of the secant line through the points P(9, g(9)) and Q(x, g(x)) from the following: a. b. c. 41. What would you have to do to your answer in the previous problem to make it the slope of the tangent line to at P(9, g(9))? 42. A tank holds 800 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. t (min) V (gal) (a) Find the average rates at which water flows from the tank (slopes of secant lines) for the time interval [10, 15]. gal/min (b) Find the average rates at which water flows from the tank (slopes of secant lines) for the time interval [15, 20]. gal/min

7 (c) The slope of the tangent line at the point (15, 208) represents the rate at which water is flowing from the tank after 15 minutes. Estimate this rate by averaging the slopes of the secant lines in parts (a) and (b). gal/min 43. One hundred dollars is deposited in a savings account at 6% interest compounded continuously. The function defined by f(x) shown in the figure gives the balance in the account after t years. At what rate (in dollars per year) is the balance growing after 15 years. 44. Refer to the figure, where f(t) is the interest rate (as a percent) on a 6-month certificate of deposit t years after January 1, The straight lines are tangent to the graph of y = f(t) at t = 2, t= 4, and t = 10. How fast was the interest rate changing on January 1, 1995

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