Textbook Question
Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.
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Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 65a
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
a. Is this function one-to-one on the interval 0 ≤ t ≤ 4?
Verified step by step guidance1
Step 1: Understand the function given: \( h(t) = 64t - 16t^2 \). This is a quadratic function representing the height of a baseball over time.
Step 2: Recall that a function is one-to-one if it passes the horizontal line test, meaning no horizontal line intersects the graph of the function more than once.
Step 3: Analyze the function \( h(t) = 64t - 16t^2 \). This is a downward-opening parabola because the coefficient of \( t^2 \) is negative.
Step 4: Determine the vertex of the parabola, which is the maximum point, using the formula \( t = -\frac{b}{2a} \) where \( a = -16 \) and \( b = 64 \).
Step 5: Evaluate whether the function is increasing or decreasing on the interval \( 0 \leq t \leq 4 \) by checking the behavior of the function before and after the vertex within this interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
One-to-One Function
A function is considered one-to-one if it assigns distinct outputs to distinct inputs, meaning that no two different inputs produce the same output. To determine if a function is one-to-one, we can use the horizontal line test: if any horizontal line intersects the graph of the function more than once, the function is not one-to-one. In the context of the given height function, analyzing its behavior over the specified interval is crucial.
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One-Sided Limits
Quadratic Functions
The function given, h(t) = 64t - 16t², is a quadratic function, which typically has a parabolic shape. Quadratic functions can open upwards or downwards depending on the sign of the leading coefficient. In this case, since the coefficient of t² is negative, the parabola opens downwards, indicating that the function will reach a maximum height before decreasing, which is important for understanding its behavior over the interval.
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6:04Introduction to Polynomial Functions
Critical Points and Intervals
Critical points of a function occur where its derivative is zero or undefined, indicating potential local maxima or minima. For the height function, finding the derivative and setting it to zero will help identify critical points within the interval [0, 4]. Analyzing these points will reveal whether the function is increasing or decreasing, which is essential for determining if it is one-to-one on the specified interval.
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04:50Critical Points
Related Practice
Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
d. At what time is the ball at a height of 30 ft on the way up?
Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = ƒ⁻¹ (h)
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Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)
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Textbook Question
Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 4x-3
Textbook Question
Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 10