#FUNDAMENTAL THEOREM OF CALCULUS PART 1 FREE#

Her terminal velocity in this position is 220 ft/sec. On Julie’s second jump of the day, she decides she wants to fall a little faster and orients herself in the “head down” position.

Find the total time Julie spends in the air, from the time she leaves the airplane until the time her feet touch the ground. After her canopy is fully open, her speed is reduced to 16 ft/sec.

It takes 5 sec for her parachute to open completely and for her to slow down, during which time she falls another 400 ft.

#FUNDAMENTAL THEOREM OF CALCULUS PART 1 FREE#

If Julie pulls her ripcord at an altitude of 3000 ft, how long does she spend in a free fall?.Based on your answer to question 1, set up an expression involving one or more integrals that represents the distance Julie falls after 30 sec.How long after she exits the aircraft does Julie reach terminal velocity?.Using this information, answer the following questions. On her first jump of the day, Julie orients herself in the slower “belly down” position (terminal velocity is 176 ft/sec). After she reaches terminal velocity, her speed remains constant until she pulls her ripcord and slows down to land. She continues to accelerate according to this velocity function until she reaches terminal velocity. After she exits the aircraft, she immediately starts falling at a velocity given by v(t)=32t. Julie executes her jumps from an altitude of 12,500 ft. Since Julie will be moving (falling) in a downward direction, we assume the downward direction is positive to simplify our calculations. If, instead, she orients her body with her head straight down, she falls faster, reaching a terminal velocity of 150 mph (220 ft/sec). If she arches her back and points her belly toward the ground, she reaches a terminal velocity of approximately 120 mph (176 ft/sec).

She has more than 300 jumps under her belt and has mastered the art of making adjustments to her body position in the air to control how fast she falls. Skydivers can adjust the velocity of their dive by changing the position of their body during the free fall. For example, if this were a profit function, a negative number indicates the company is operating at a loss over the given interval.įigure 5. Area is always positive, but a definite integral can still produce a negative number (a net signed area). Note that the region between the curve and the x-axis is all below the x-axis. The region of the area we just calculated is depicted in Figure 3. This always happens when evaluating a definite integral. If we had chosen another antiderivative, the constant term would have canceled out. So, for convenience, we chose the antiderivative with C=0. The reason is that, according to the Fundamental Theorem of Calculus, Part 2, any antiderivative works. Notice that we did not include the “+ C” term when we wrote the antiderivative. Then check your work with your calculator.F(x)= The function F is called an antiderivative of the function f.ġ6.3.5 Use the corollary to predict the value ofġ6.3.6 Use the corollary of the Fundamental theorem to evaluate Hint: You will have to use the chain rule.ġ6.3.3 Evaluate the following derivative and check with your TI-89.ġ6.3.4 Find a more general version of the Fundamental theorem by predicting the following derivative. The function named F is the same as the area function that was previously explored.ġ6.3.1 Use the restatement of the Fundament theorem to evaluate the following derivatives, then check your predictions with the TI-89.ġ6.3.2 Predict the following derivative. This is true regardless of the value of the lower limit a. That is, the derivative of a definite integral of f whose upper limit is the variable x and whose lower limit is the constant a equals the function f evaluated at x. Has a derivative at every point in, and the derivative is If f is continuous on, then the function The following is a restatement of the Fundamental Theorem. You have discovered the Fundamental Theorem in the context of finding areas under a curve but a more general version of this theorem can be proved without an appeal to area. A restatement of the Fundamental Theorem of Calculus is presented in this lesson along with a corollary that is used to find the value of a definite integral analytically.

Fundamental theorem of calculus part 1