please write out and show work for each question answered.
1) 6 pts using direct substitution, evaluate the following limits.
a. $ \lim_{x \to 2} x^3 - 6x + 3 = $
b. $ \lim_{x \to 1} \frac{x^2 + 1}{3x^2} = $
2) (9 pts) find the derivative of the following functions using power rule (assumed context):
a. $ f(x) = 5x^{10} + 2 $
$ f(x) = \underline{\quad} $
b. $ g(x) = 4x^{3/2} $
$ g(x) = \underline{\quad} $
c. $ h(x) = \frac{3}{x^2} $
$ h(x) = \underline{\quad} $
3) (3 pts) suppose that $ f(5) = 2, f(5) = 4, g(5) = -2, $ and $ g(5) = -1 $.
find $ h(5) $ where $ h(x) = 3f(x) - 5g(x) $
$ h(5) = \underline{\quad} $

Question

please write out and show work for each question answered.
1) 6 pts using direct substitution, evaluate the following limits.
  a. $ \lim_{x \to 2} x^3 - 6x + 3 = $
  b. $ \lim_{x \to 1} \frac{x^2 + 1}{3x^2} = $
2) (9 pts) find the derivative of the following functions using power rule (assumed context):
  a. $ f(x) = 5x^{10} + 2 $
    $ f(x) = \underline{\quad} $
  b. $ g(x) = 4x^{3/2} $
    $ g(x) = \underline{\quad} $
  c. $ h(x) = \frac{3}{x^2} $
    $ h(x) = \underline{\quad} $
3) (3 pts) suppose that $ f(5) = 2, f(5) = 4, g(5) = -2, $ and $ g(5) = -1 $.
  find $ h(5) $ where $ h(x) = 3f(x) - 5g(x) $
  $ h(5) = \underline{\quad} $

Accepted Answer

### Problem 1a
# Explanation:
## Step1: Substitute $ x = 2 $ into the function
$ \lim_{x
ightarrow 2} x^{3}-6x + 3=2^{3}-6\times2 + 3 $
## Step2: Calculate each term
$ 2^{3}=8 $, $ 6\times2 = 12 $, so $ 8-12 + 3=-1 $
# Answer:
$ -1 $

### Problem 1b
# Explanation:
## Step1: Substitute $ x = 1 $ into the function
$ \lim_{x
ightarrow 1}\frac{x^{2}+1}{3x^{2}}=\frac{1^{2}+1}{3\times1^{2}} $
## Step2: Simplify the numerator and denominator
Numerator: $ 1 + 1=2 $, Denominator: $ 3\times1 = 3 $, so $ \frac{2}{3} $
# Answer:
$ \frac{2}{3} $

### Problem 2a
# Explanation:
## Step1: Apply the power rule $ \frac{d}{dx}(x^{n})=nx^{n - 1} $ and constant rule (derivative of constant is 0)
For $ f(x)=5x^{10}+2 $, derivative of $ 5x^{10} $ is $ 5\times10x^{10 - 1} $, derivative of 2 is 0.
## Step2: Simplify
$ 5\times10x^{9}=50x^{9} $
# Answer:
$ 50x^{9} $

### Problem 2b
# Explanation:
## Step1: Apply the power rule $ \frac{d}{dx}(x^{n})=nx^{n - 1} $
For $ g(x)=4x^{\frac{3}{2}} $, derivative is $ 4\times\frac{3}{2}x^{\frac{3}{2}-1} $
## Step2: Simplify
$ 4\times\frac{3}{2}=6 $, $ \frac{3}{2}-1=\frac{1}{2} $, so $ 6x^{\frac{1}{2}} $
# Answer:
$ 6x^{\frac{1}{2}} $ (or $ 6\sqrt{x} $)

### Problem 2c
# Explanation:
## Step1: Rewrite the function as $ h(x)=3x^{-2} $
## Step2: Apply the power rule $ \frac{d}{dx}(x^{n})=nx^{n - 1} $
Derivative is $ 3\times(-2)x^{-2-1} $
## Step3: Simplify
$ 3\times(-2)=-6 $, $ -2 - 1=-3 $, so $ -6x^{-3}=\frac{-6}{x^{3}} $
# Answer:
$ -6x^{-3} $ (or $ \frac{-6}{x^{3}} $)

### Problem 3
# Explanation:
## Step1: Use the sum/difference rule and constant multiple rule for derivatives. If $ h(x)=3f(x)-5g(x) $, then $ h'(x)=3f'(x)-5g'(x) $
## Step2: Substitute $ x = 5 $
$ h'(5)=3f'(5)-5g'(5) $
## Step3: Plug in the given values $ f'(5) = 4 $, $ g'(5)=-1 $
$ 3\times4-5\times(-1)=12 + 5=17 $
# Answer:
$ 17 $

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QUESTION IMAGE

Question

Explanation:

Problem 1a

Step1: Substitute \( x = 2 \) into the function

Step2: Calculate each term

Step1: Substitute \( x = 1 \) into the function

Step2: Simplify the numerator and denominator

Step1: Apply the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \) and constant rule (derivative of constant is 0)

Step2: Simplify

Answer:

Problem 1b