Textbook Question
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 3x2 - 4x; P(1, -1)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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 3, Problem 3.1.24a
Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.
f(x) = 8 - 2x2; P(0, 8)
Verified step by step guidance1
Identify the function f(x) = 8 - 2x^2 and the point P(0, 8) where you need to find the slope of the tangent line.
Recall the definition of the derivative as the slope of the tangent line at a point: f'(x) = lim_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).
Substitute f(x) = 8 - 2x^2 into the derivative definition: f'(x) = lim_{h \(\to\) 0} \(\frac{(8 - 2(x+h)^2) - (8 - 2x^2)}{h}\).
Simplify the expression inside the limit: f'(x) = lim_{h \(\to\) 0} \(\frac{-2(x^2 + 2xh + h^2) + 2x^2}{h}\).
Further simplify and evaluate the limit as h approaches 0 to find f'(x), then substitute x = 0 to find the slope of the tangent line at P(0, 8).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line represents the instantaneous rate of change of the function at that point, which is crucial for understanding how the function behaves locally.
Recommended video:
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function at a point quantifies how the function's output changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this context, finding the derivative of f(x) will provide the slope of the tangent line at point P.
Recommended video:
05:44Derivatives
Limit Definition of Derivative
The limit definition of the derivative states that the derivative f'(a) at a point a is the limit of the difference quotient as h approaches zero: f'(a) = lim(h→0) [(f(a+h) - f(a))/h]. This definition is fundamental for calculating the slope of the tangent line using the function's values around the point of interest.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(s) = 4s³+3s; a= -3, -1
Textbook Question
Consider the following cost functions.
a. Find the average cost and marginal cost functions.
C(x) = 1000+0.1x, 0≤x≤5000, a=2000
Textbook Question
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
(x+y)^2/3=y; (4, 4)
Textbook Question
Analyzing slopes Use the points A, B, C, D, and E in the following graphs to answer these questions. <IMAGE>
a. At which points is the slope of the curve negative?
2
views
Textbook Question
The following table gives the distance f(t) fallen by a smoke jumper seconds after she opens her chute. <IMAGE>
a. Use the forward difference quotient with ℎ = 0.5 to estimate the velocity of the smoke jumper at t=2 seconds.