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문제 57
Find and simplify the derivative of the following functions.
h(x) = (5x7 + 5x)(6x3 + 3x2 + 3)
문제 58
Calculate the derivative of the following functions.
y = (1 - e0.05x)-1
문제 59
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.
문제 59a
{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
문제 59b
{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
문제 60
Calculate the derivative of the following functions.
y = √x+√x+√x
문제 61a
{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = ex; a = ln 3
문제 61b
{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 = e^x; a = ln 3
문제 62
Calculate the derivative of the following functions.
y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants
문제 62a
Find an equation of the line tangent to the given curve at a.
y = 2x2 / (3x - 1); a = 1
문제 62b
Use a graphing utility to graph the curve and the tangent line on the same set of axes.
y = 2x2 / (3x - 1); a = 1
문제 63
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y=3x−4; P(1, −1)
문제 63a
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 0.
문제 63b
Let f(x) = x2 - 6x + 5.
Find the values of x for which the slope of the curve y = f(x) is 2.
문제 64
Calculate the derivative of the following functions.
y = (e^x / x+1)⁸
문제 65
A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.
y = 2/x; P(1, 2)
문제 66
Calculate the derivative of the following functions.
y = tan(xe^x)
문제 66a
Population growth Consider the following population functions.
a. Find the instantaneous growth rate of the population, for t≥0.
p(t) = 600 (t²+3/t²+9)
문제 66b
Population growth Consider the following population functions.
b. What is the instantaneous growth rate at t=5?
p(t) = 600 (t²+3/t²+9)
문제 66c
Population growth Consider the following population functions.
c. Estimate the time when the instantaneous growth rate is greatest.
p(t) = 600 (t²+3/t²+9)
문제 66d
{Use of Tech} Population growth Consider the following population functions.
d. Evaluate and interpret lim t→∞ p(t).
p(t) = 600 (t²+3/t²+9)
문제 66e
Population growth Consider the following population functions.
e.Use a graphing utility to graph the population and its growth rate.
p(t) = 600 (t²+3/t²+9)
문제 67
Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x)=x²+1; Q(3, 6)
문제 67a
Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line is horizontal.
문제 67b
Let f(x) = 4√x - x.
Find all points on the graph of f at which the tangent line has slope -1/2.
문제 68
Find f′(x), f′′(x), and f′′′(x) for the following functions.
f(x) = 3x3 + 5x2 + 6x
문제 69
Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.
f(x) = 1/x; Q (-2, 4)
문제 69a
The following equations implicitly define one or more functions.
a. Find dy/dx using implicit differentiation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
문제 69b
The following equations implicitly define one or more functions.
b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)
문제 69c
The following equations implicitly define one or more functions.
c. Use the functions found in part (b) to graph the given equation.
x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)