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Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.
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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 58
Calculate the derivative of the following functions.
y = (1 - e0.05x)-1
Guida verificata passo dopo passo1
Step 1: Identify the function y = (1 - e^{0.05x})^{-1} as a composition of functions, where the outer function is u^{-1} and the inner function is u = 1 - e^{0.05x}.
Step 2: Apply the chain rule for differentiation, which states that the derivative of a composite function y = f(g(x)) is given by y' = f'(g(x)) * g'(x).
Step 3: Differentiate the outer function u^{-1} with respect to u, which gives -u^{-2}.
Step 4: Differentiate the inner function u = 1 - e^{0.05x} with respect to x, which gives -0.05e^{0.05x}.
Step 5: Combine the results from Steps 3 and 4 using the chain rule: y' = -u^{-2} * (-0.05e^{0.05x}), and substitute u = 1 - e^{0.05x} back into the expression.
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Derivative
The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule, depending on the form of the function.
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05:44
Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative dy/dx can be found by multiplying the derivative of the outer function f with respect to g by the derivative of the inner function g with respect to x. This rule is essential when differentiating functions that are nested within each other.
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Exponential Functions
Exponential functions are mathematical functions of the form y = a * e^(bx), where e is the base of the natural logarithm, and a and b are constants. These functions are characterized by their rapid growth or decay and are commonly encountered in calculus. The derivative of an exponential function is unique in that it is proportional to the function itself, making it crucial for solving problems involving growth rates.
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The energy (in joules) released by an earthquake of magnitude M is given by the equation E = 25,000 ⋅ 101.5M. (This equation can be solved for M to define the magnitude of a given earthquake; it is a refinement of the original Richter scale created by Charles Richter in 1935.)
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Find and simplify the derivative of the following functions.
h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)
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Domanda del libro di testo
{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = −3x2 + 2; a=1
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The power is the rate of energy consumption; that is, P(t) = E′(t) Find the power over the interval 0 ≤ t ≤ 24.
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Domanda del libro di testo
{Use of Tech} Equations of tangent lines
b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = −3x²+2; a=1
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