Domanda del libro di testo
Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>
b. (f^-1)'(6)
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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 3.2.30b
21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(t) = 3t⁴; a= -2, 2
Guida verificata passo dopo passo1
Step 1: Identify the function f(t) = 3t^4 and the values of a for which we need to evaluate the derivative, which are a = -2 and a = 2.
Step 2: Find the derivative of the function f(t) with respect to t. Use the power rule for differentiation, which states that if f(t) = t^n, then f'(t) = n*t^(n-1).
Step 3: Apply the power rule to f(t) = 3t^4. The derivative f'(t) is found by multiplying the exponent by the coefficient and reducing the exponent by one.
Step 4: Substitute the given values of a into the derivative f'(t) to find f'(-2) and f'(2).
Step 5: Simplify the expressions obtained from substituting a = -2 and a = 2 into f'(t) to find the values of the derivative at these points.
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Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.
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Percorso guidato
05:44
Derivatives
Power Rule
The power rule is a basic differentiation rule used to find the derivative of functions of the form f(t) = t^n, where n is a real number. According to this rule, the derivative f'(t) is given by n*t^(n-1). This rule simplifies the process of differentiation, especially for polynomial functions, making it easier to evaluate derivatives at specific points.
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5:50Power Rules
Evaluating Derivatives at Specific Points
Evaluating the derivative at specific points involves substituting the given values into the derivative function. For instance, once the derivative f'(t) is calculated, substituting a = -2 or a = 2 into f'(t) yields the slope of the tangent line at those points on the original function. This process is crucial for understanding the behavior of the function at specific locations.
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04:50Critical Points
Pratica correlata
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Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
b. Find equations of all lines tangent to the curve y(x²+4)=8 when y=1.
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Domanda del libro di testo
Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = (x + 5) / (x - 1); a = 3
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Domanda del libro di testo
79–82. {Use of Tech} Visualizing tangent and normal lines
b. Graph the tangent and normal lines on the given graph.
(x²+y²)² = 25/3 (x²-y²); (x0,y0) = (2,-1) (lemniscate of Bernoulli)
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Domanda del libro di testo
{Use of Tech} Computing limits with angles in degrees Suppose your graphing calculator has two functions, one called sin x, which calculates the sine of x when x is in radians, and the other called s(x), which calculates the sine of x when x is in degrees.
b. Evaluate lim x→0 s(x) / x. Verify your answer by estimating the limit on your calculator.
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Domanda del libro di testo
45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
(x²+y²)²=25/4 xy²; (1, 2)
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