48. Evaluate [tex]$\lim _{x \rightarrow \infty}\left(1-\frac{3}{x}\right)^x$[/tex].
A. 3
B. [tex]$r^3$[/tex]
C. [tex]$e^{-3}$[/tex]
D. -3 .
49. If [tex]$x=t^2+1$[/tex] and [tex]$u=t^3+2$[/tex], tind [tex]$\frac{d y}{d x}$[/tex]. A. [tex]$2 t$[/tex]
B. [tex]$34 C \cdot \frac{3}{2}$[/tex]
D. [tex]$\frac{4}{3}$[/tex]
50. If [tex]$y=(3 x+8)^{19}$[/tex]. obtain [tex]$y^{\prime}$[/tex]
A. [tex]$(3 x+8)^{18}$[/tex]
B. [tex]$27(3 x$[/tex] D. [tex]$19(3 x+8)^{18}$[/tex]
51. Find [tex]$\frac{d y}{d x}$[/tex] if [tex]$y=\cos ^2 x+\sin ^2 x$[/tex], A. [tex]$-2 \sin x \cos x$[/tex]
B. 0
C. 2 cm
52. Find [tex]$\frac{d y}{d x}$[/tex] if [tex]$3+y^2=x^3-u$[/tex].
A. [tex]$\frac{3 y^2}{2 y+1}$[/tex]
B. [tex]$\frac{3 x^2}{2 y-1}$[/tex]
C. [tex]$\frac{x^2}{2 y^2+1}$[/tex]
D. [tex]$\frac{x}{24}$[/tex]
53. Given [tex]$y(x)=3 r^{2 x}+\sin x$[/tex], find [tex]$y^{\prime \prime}(\theta)$[/tex].
A. 3 B.
12
C.GD.
54. Given [tex]$y=6 . x^{-i x}$[/tex], find [tex]$\frac{d y}{d x}$[/tex]
A. [tex]$6 e^{-5 x}(1-5 x)$[/tex]
B. [tex]$x^{-5 x} 16 x$[/tex] D. [tex]$\left(i e^{-5 x}(1+5 x)\right.$[/tex]
55. Find the derivative of [tex]$y=\frac{x^2-1}{2 x+1}$[/tex] at [tex]$x=0$[/tex].
A. [tex]$\frac{1}{4}$[/tex]
B. [tex]$\frac{1}{3}$[/tex]
56. Find the derivative of [tex]$y=\ln \left(x^3\right)+\sin x$[/tex]
A. [tex]$\frac{3}{x}$[/tex]
Answer (1)
Problem 48 uses the limit definition to find lim x → ∞ ( 1 − x 3 ) x = e − 3 .
Problem 49 applies the chain rule to find d x d y = 2 3 t given x = t 2 + 1 and y = t 3 + 2 .
Problem 50 uses the chain rule to obtain y ′ = 57 ( 3 x + 8 ) 18 for y = ( 3 x + 8 ) 19 .
Problem 51 simplifies y = cos 2 x + sin 2 x to y = 1 , thus d x d y = 0 .
Problem 52 uses implicit differentiation to find d x d y = 2 y 3 x 2 from 3 + y 2 = x 3 − u , assuming u is constant.
Problem 53 calculates the second derivative y ′′ ( θ ) = 12 e 2 θ − sin θ for y ( x ) = 3 e 2 x + sin x .
Problem 54 finds the derivative d x d y = − 30 e − 5 x for y = 6 e − 5 x .
Problem 55 uses the quotient rule to find the derivative of y = 2 x + 1 x 2 − 1 at x = 0 , resulting in 2.
Problem 56 determines the derivative of y = ln ( x 3 ) + sin x as d x d y = x 3 + cos x .
The final answers are: e − 3 , 2 3 t , 57 ( 3 x + 8 ) 18 , 0 , 2 y 3 x 2 , 12 e 2 θ − sin θ , − 30 e − 5 x , 2 , x 3 + cos x
Explanation
Introduction We are given a set of calculus problems to solve, including finding limits and derivatives. We will address each problem individually, showing the steps and reasoning for each solution.
Problem 48 Solution Problem 48: Evaluate lim x → ∞ ( 1 − x 3 ) x . This is a limit of the form lim x → ∞ ( 1 + x a ) x = e a . In this case, a = − 3 . Therefore, the limit is e − 3 .
Problem 49 Solution Problem 49: If x = t 2 + 1 and y = t 3 + 2 , find d x d y . We use the chain rule: d x d y = d x / d t d y / d t . First, we find d t d x = 2 t and d t d y = 3 t 2 . Then, d x d y = 2 t 3 t 2 = 2 3 t .
Problem 50 Solution Problem 50: If y = ( 3 x + 8 ) 19 , obtain y ′ . We use the chain rule: y ′ = 19 ( 3 x + 8 ) 18 ⋅ 3 = 57 ( 3 x + 8 ) 18 .
Problem 51 Solution Problem 51: Find d x d y if y = cos 2 x + sin 2 x . Since cos 2 x + sin 2 x = 1 , we have y = 1 . Therefore, d x d y = 0 .
Problem 52 Solution Problem 52: Find d x d y if 3 + y 2 = x 3 − u . We use implicit differentiation with respect to x . Assuming u is a function of x , we have 2 y d x d y = 3 x 2 − d x d u . If we assume u is a constant, then d x d u = 0 , so 2 y d x d y = 3 x 2 , which gives d x d y = 2 y 3 x 2 .
Problem 53 Solution Problem 53: Given y ( x ) = 3 e 2 x + sin x , find y ′′ ( θ ) . First, we find y ′ ( x ) = 6 e 2 x + cos x . Then, we find the second derivative y ′′ ( x ) = 12 e 2 x − sin x . Evaluating at x = θ , we get y ′′ ( θ ) = 12 e 2 θ − sin θ .
Problem 54 Solution Problem 54: Given y = 6 e − 5 x , find d x d y . We have d x d y = 6 ( − 5 ) e − 5 x = − 30 e − 5 x .
Problem 55 Solution Problem 55: Find the derivative of y = 2 x + 1 x 2 − 1 at x = 0 . We use the quotient rule: d x d y = ( 2 x + 1 ) 2 ( 2 x + 1 ) ( 2 x ) − ( x 2 − 1 ) ( 2 ) = ( 2 x + 1 ) 2 4 x 2 + 2 x − 2 x 2 + 2 = ( 2 x + 1 ) 2 2 x 2 + 2 x + 2 . Evaluating at x = 0 , we get d x d y = 1 2 = 2 .
Problem 56 Solution Problem 56: Find the derivative of y = ln ( x 3 ) + sin x . We can rewrite y = 3 ln ( x ) + sin x . Then, d x d y = x 3 + cos x .
Final Answers The solutions to the problems are:
C. e − 3
C. 2 3 t
B. 57 ( 3 x + 8 ) 18
B. 0
B. 2 y 3 x 2
D. 12 e 2 θ − sin θ
A. − 30 e − 5 x
C. 2
C. x 3 + cos x
Examples
Understanding limits is crucial in physics, especially when dealing with asymptotic behavior. For instance, when analyzing the motion of an object approaching the speed of light, we use limits to describe how its kinetic energy increases without bound as its velocity approaches c . Similarly, derivatives are fundamental in economics for marginal analysis, where they help determine the change in cost or revenue resulting from a small change in production or sales. These calculus concepts provide essential tools for modeling and optimizing real-world phenomena.
