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Math 11-Worksheet 04 - worksheet

worksheet
Course

Calculus I (MATH 011)

81 Documents
Students shared 81 documents in this course
Academic year: 2022/2023
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This document has been uploaded by a student, just like you, who decided to remain anonymous.
University of California, Merced

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1 Section 3.

  1. At time t in seconds, a particle’s distance s(t), in micrometers (μm), from a point is given by

s(t) = 3 4 t 2 − 1

What is the average velocity of the particle from t = 2 to t = 4? 2. Consider the graph below of the position of a model rocket, h(t). Here h is in meters and t is measured in seconds.

(a) Estimate the average velocity of the rocket in the first 3 seconds of flight. (b) What is the average velocity of the rocket in the last 3 seconds of flight? (c) Estimate the average velocity of the rocket between 1 and 3 seconds. (d) Estimate the average velocity of the rocket in the last 5 seconds of flight. (e) In general, under what conditions is the average velocity for the rocket zero? 3. For each of the following situations, sketch a graph of the distance the car has traveled as a function of time. (a) A car is driven at a constant speed. (b) A car is driven at an increasing speed. (c) A car starts at a high speed, and its speed then decreases slowly.

  1. Write out and simplify the difference quotient f (x + h h) − f (x)for each of the following functions. (a) f (x) = 2x 2 + 3 (b) g(x) = 4 x

  2. Estimate the instantaneous velocity at the points x = − 1 , 0 , and 1 for the given position function in the figure below.

  3. Compute the instantaneous velocity (i. slope of the tangent line), using the limit definition, for the given position functions at the indicated values. (a) s(t) = 3 − 4 t 2 for t = 1 (b) y(x) = 1 2 x − 1 for x = 10 (c) f (w) = w 3 for w = − 2

  4. Write the equation of the tangent line to the function at the indicated point.

(a) f (x) = 1 − 3 x at x = 3 (b) g(x) = x 2 at x = − 1 (c) h(x) = − x 2 at x = 2

  1. Calculate the derivative of each function at the given point.

(a) A(x) = 1 2 x 2 + 1 at x = 3

(b) B(x) = x 12 at x = − 2 (c) C(x) =

√

x at x = 4 Hint: Rationalize the numerator.

  1. A model rocket follows the trajectory given by h(t) = − 16 t 2 + 120t + 250, with t in seconds and h in feet. (a) Find the instantaneous velocity, v(t). (b) Does the rocket ever stop? If so, at what height? (c) Calculate the instantaneous velocity after 2 seconds and after 6 seconds. What do you think the significance of the signs are?
  2. A pool is being filled with water and is currently half full. The volume, measured in cubic feet, of the pool after t minutes is described by the function V (t) = √ 2 t + 1. Find the instantaneous rate of change of the volume of the pool, and calculate this volume for when 12 minutes have passed.
  3. A aircraft approaching a runway and landing follows the flightpath given (in meters) by y(t) = t 50 + 1 , with t in seconds. Find the instantaneous velocity for the plane, and use it to calculate the velocity after 3 seconds.
  4. A chemical reaction combines two products, x (in milligrams), producing a current (in milliamps) according to the function f (x) = 2 + x 12. Find the instantaneous rate of change for the current produced when using 2 milligrams of the product.

2 Section 3

  1. Find the intervals where f (x), given in the figure below, is (a) increasing and (b) decreasing.

  2. Sketch a graph of the derivative of the given functions given in the figures below.

  3. Consider the function, f (x), in the figure given below over the interval [− 5 , 5].

(a) On what intervals is f (x) continuous? (b) If there are discontinuities, what kind are they? (c) Where is f (x) (i) increasing, (ii) decreasing, and (iii) neither? (d) On what intervals is f (x) differentiable?

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Math 11-Worksheet 04 - worksheet

Course: Calculus I (MATH 011)

81 Documents
Students shared 81 documents in this course

University: University of California, Merced

Was this document helpful?
Math 11 Worksheet 4 - Sections 3.1 and 3.2a Spring 2023
1 Section 3.1
1. At time tin seconds, a particle’s distance s(t), in micrometers (µm), from a point is given by
s(t) = 3
4t2−1
What is the average velocity of the particle from t= 2 to t= 4?
2. Consider the graph below of the position of a model rocket, h(t). Here his in meters and tis measured
in seconds.
(a) Estimate the average velocity of the rocket in the first 3 seconds of flight.
(b) What is the average velocity of the rocket in the last 3 seconds of flight?
(c) Estimate the average velocity of the rocket between 1 and 3 seconds.
(d) Estimate the average velocity of the rocket in the last 5 seconds of flight.
(e) In general, under what conditions is the average velocity for the rocket zero?
3. For each of the following situations, sketch a graph of the distance the car has traveled as a function
of time.
(a) A car is driven at a constant speed.
(b) A car is driven at an increasing speed.
(c) A car starts at a high speed, and its speed then decreases slowly.
4. Write out and simplify the difference quotient f(x+h)−f(x)
hfor each of the following functions.
(a) f(x) = 2x2+ 3 (b) g(x) = 4
x
1

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