Textbook Question
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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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 58
Calculate the derivative of the following functions.
y = (1 - e0.05x)-1
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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Related Practice
Textbook Question
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.)
Compute the energy released by earthquakes of magnitude 1, 2, 3, 4, and 5. Plot the points on a graph and join them with a smooth curve.
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Textbook Question
Find and simplify the derivative of the following functions.
h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)
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Textbook Question
{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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Textbook Question
Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal=4184 J. One hour of walking consumes roughly 10⁶J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (1 kW=10³ W) and megawatts (1 MW=10⁶ W). If energy is used at a rate of 1 kW for one hour, the total amount of energy used is 1 kilowatt-hour (1 kWh = 3.6×10⁶ J) Suppose the cumulative energy used in a large building over a 24-hr period is given by E(t)=100t + 4t² − (t³ / 9) kWh where t = 0 corresponds to midnight.
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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Textbook Question
{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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