Calculus: 1001 Practice Problems For Dummies (+ Free Online Practice). Patrick Jones

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Using the Second Derivative Test to Find Local Maxima and Minima

       476–480 Use the second derivative test to find the local maxima and local minima of the given function.

      476.

477.

      478.

      479.

      480. on

Applying Rolle’s Theorem

       481– 483 Verify that the function satisfies the hypotheses of Rolle’s theorem. Then find all values c in the given interval that satisfy the conclusion of Rolle’s theorem.

      481. ,

      482. , [–8, 0]

      483. ,

Using the Mean Value Theorem

       484–486 Verify that the given function satisfies the hypotheses of the mean value theorem. Then find all numbers c that satisfy the conclusion of the mean value theorem.

      484. ,

      485. ,

      486. ,

Applying the Mean Value Theorem to Solve Problems

       487– 489 Solve the problem related to the mean value theorem.

487. If and for , what is the smallest possible value of ? Assume that f satisfies the hypothesis of the mean value theorem.

      488. Suppose that for all values of x. What are the strictest bounds you can put on the value of ? Assume that f is differentiable for all x.

      489. Apply the mean value theorem to the function on the interval to find bounds for the value of .

Relating Velocity and Position

       490–492 Use the position function s(t) to find the velocity and acceleration at the given value of t. Recall that velocity is the change in position with respect to time and acceleration is the change in velocity with respect to time.

      490. at

      491. at

      492. at

Finding Velocity and Speed

       493– 497 Solve the given question related to speed or velocity. Recall that velocity is the change in position with respect to time.

      493. A mass on a spring vibrates horizontally with an equation of motion given by , where x is measured in feet and t is measured in seconds. Is the spring stretching or compressing at ? What is the speed of the spring at that time?

      494. A stone is thrown straight up with the height given by the function , where s is measured in feet and t is measured in seconds. What is the maximum height of the stone? What is the velocity of the stone when it’s 20 feet above the ground on its way up? And what is its velocity at that height on the way down? Give exact answers.

495. A stone is thrown vertically upward with the height given by , where s is measured in feet and t is measured in seconds. What is the maximum height of the stone? What is the velocity of the stone when it hits the ground?

      496. A particle moves on a vertical line so that its coordinate at time t is given by for . When is the particle moving upward, and when is it moving downward? Give an exact answer in interval notation.

      497. СКАЧАТЬ