Textbook Question
Calculate the derivative of the following functions.
y = csc ex
Ch. 3 - Derivatives
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 3, Problem 34b
Derivatives and tangent lines
b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.
f(x) = x²; a=3
Verified step by step guidance1
Step 1: Find the derivative of the function f(x) = x^2. The derivative, f'(x), represents the slope of the tangent line at any point x on the graph of f.
Step 2: Calculate the derivative f'(x) = 2x. This is done by applying the power rule, which states that the derivative of x^n is n*x^(n-1).
Step 3: Evaluate the derivative at the given point a = 3 to find the slope of the tangent line. Substitute x = 3 into f'(x) to get f'(3) = 2*3.
Step 4: Determine the y-coordinate of the point on the graph by evaluating f(a). Substitute x = 3 into f(x) to get f(3) = 3^2.
Step 5: Use the point-slope form of a line, y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point (3, f(3)), to write the equation of the tangent line.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative
The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = x², the derivative f'(x) can be calculated using the power rule, which states that the derivative of x^n is n*x^(n-1).
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Derivatives
Tangent Line
A tangent line to a curve at a given point is a straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line can be expressed in point-slope form: y - f(a) = f'(a)(x - a), where (a, f(a)) is the point of tangency and f'(a) is the derivative at that point.
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Point-Slope Form
Point-slope form is a way to express the equation of a line using a specific point on the line and its slope. The general formula is y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. This form is particularly useful for writing the equation of a tangent line once the slope and point of tangency are known.
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Related Practice
Textbook Question
Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = x²; a=3
Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.
Textbook Question
Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.
For what prices is the demand elastic? Inelastic?
Textbook Question
Find the derivative function f' for the following functions f.
f(x) =3x²+2x−10; a=1
Textbook Question
Calculate the derivative of the following functions.
y = tan ex