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Problem 3.1.20
In Exercises 19–22, find the slope of the curve at the point indicated.
y = x³ − 2x + 7, x = −2
Problem 3.9.49
The diameter of a sphere is measured as 100 ± 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.
Problem 3.9.38
Differential Estimates of Change
In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.
The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change
Problem 3.33
Find the derivatives of the functions in Exercises 1–42.
_____
𝔂 = / x² + x
√ x²
Problem 3.87
In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.
__
x + √xy = 6, (4, 1)
Problem 3.5.49
Theory and Examples
The equations in Exercises 49 and 50 give the position s = f(t) of a body moving on a coordinate line (s in meters, t in seconds). Find the body’s velocity, speed, acceleration, and jerk at time t = π/4 sec.
s = 2 − 2 sin t
Problem 3.1.43
[Technology Exercise]
Graph the curves in Exercises 39–48.
a. Where do the graphs appear to have vertical tangent lines?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.
y = 4x²/⁵ − 2x
Problem 3.3.9
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = 6x² − 10x − 5x⁻²
Problem 3.3.2
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x² + x + 8
Problem 3.3.59
The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).
Problem 3.5.29
Derivatives
In Exercises 27–32, find dp/dq.
p = (sin q + cos q) / cos q
Problem 3.7.30
In Exercises 29 and 30, find the slope of the curve at the given points.
(x² + y²)² = (x – y)² at (1,0) and (1,–1)
Problem 3.7.39
Slopes, Tangent Lines, and Normal Lines
In Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.
y = 2 sin(πx – y), (1,0)
Problem 3.1.18
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
f(x) = √(x + 1), (8, 3)
Problem 3.3.65
Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form
P = (nRT / (V − nb)) − (an² / V²),
in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)
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Problem 3.2.39
One-Sided Derivatives
Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.
Problem 3.6.29
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
Problem 3.1.14
In Exercises 11–18, find the slope of the function’s graph at the given point. Then find an equation for the line tangent to the graph there.
g(x) = 8 / x², (2, 2)
Problem 3.6.33
Find the derivatives of the functions in Exercises 19–40.
f(x) = √(7 + x sec x)
Problem 3.3.28
Find the derivatives of the functions in Exercises 17–28.
y = ((x + 1)(x + 2)) / ((x − 1)(x − 2))
Problem 3.8.5
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Problem 3.3.4
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
w = 3z⁷ − 7z³ + 21z²
Problem 3.5.15
Derivatives
In Exercises 1–18, find dy/dx.
y = (sec x + tan x)(sec x − tan x)
Problem 3.6.89
Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².
Problem 3.7.22
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
y² – 2x = 1 – 2y
Problem 3.9.44
Estimating height of a building A surveyor, standing 30 ft from the base of a building, measures the angle of elevation to the top of the building to be 75°. How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%?
Problem 3.15
Find the derivatives of the functions in Exercises 1–42.
s = (sec t + tan t)⁵
Problem 3.9.18
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = x√(1 − x²)
Problem 3.6.77
Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.
Problem 3.9.20
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = (2√x)/(3(1 + √x))
