Domanda del libro di testo
Vertical tangent lines
a. Determine the points where the curve x+y³−y=1 has a vertical tangent line (see Exercise 60).
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Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.5.72a
Find an equation of the line tangent to the following curves at the given value of x.
y = 4 sin x cos x; x = π/3
Guida verificata passo dopo passo1
First, understand that the problem requires finding the equation of the tangent line to the curve y = 4 sin x cos x at x = π/3. The tangent line will have the form y = mx + b, where m is the slope at the given point and b is the y-intercept.
To find the slope of the tangent line, we need to compute the derivative of the function y = 4 sin x cos x with respect to x. Use the product rule for differentiation, which states that if you have a function u(x)v(x), its derivative is u'(x)v(x) + u(x)v'(x).
Apply the product rule: Let u(x) = 4 sin x and v(x) = cos x. Then, u'(x) = 4 cos x and v'(x) = -sin x. The derivative of y with respect to x, denoted as dy/dx, is: dy/dx = u'(x)v(x) + u(x)v'(x) = 4 cos x * cos x + 4 sin x * (-sin x).
Simplify the expression for dy/dx: dy/dx = 4 cos^2 x - 4 sin^2 x. This can be further simplified using the trigonometric identity cos(2x) = cos^2 x - sin^2 x, giving dy/dx = 4 cos(2x).
Evaluate the derivative at x = π/3 to find the slope m of the tangent line: m = 4 cos(2 * π/3). Calculate the value of y at x = π/3 using the original function y = 4 sin x cos x, and use these values to find the equation of the tangent line in the form y = mx + b.
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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 is equal to the derivative of the function at that point. This concept is crucial for finding the equation of the tangent line, as it provides both the slope and the point of tangency.
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Percorso guidato
05:13
Slopes of Tangent Lines
Derivative
The derivative of a function measures 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, calculating the derivative of the function y = 4 sin x cos x will allow us to find the slope of the tangent line at x = π/3.
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Percorso guidato
05:44Derivatives
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. In this problem, the identity sin(2x) = 2 sin x cos x can simplify the function y = 4 sin x cos x to y = 2 sin(2x), making it easier to differentiate and evaluate at specific points.
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7:17Verifying Trig Equations as Identities
Pratica correlata
Domanda del libro di testo
Airline travel The following figure shows the position function of an airliner on an out-and-back trip from Seattle to Minneapolis, where s = f(t) is the number of ground miles from Seattle t hours after take-off at 6:00 A.M. The plane returns to Seattle 8.5 hours later at 2:30 P.M. <IMAGE>
a. Calculate the average velocity of the airliner during the first 1.5 hours of the trip (0 ≤ t ≤ 1.5).
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Domanda del libro di testo
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4
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Domanda del libro di testo
The line tangent to the curve y=h(x) at x=4 is y = −3x+14. Find an equation of the line tangent to the following curves at x=4.
y = (x²-3x)h(x)
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Domanda del libro di testo
{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).
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Domanda del libro di testo
13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
sin y = 5x⁴−5; (1, π)
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