### Mathematical Analysis and Chain Rule Application **Given Functions:** 1. Let \( f(x, y) = xy^2 \) 2. Let \( \mathbf{r}(t) = \left\langle \frac{1}{2} t, 2t^3 \right\rangle \) **Tasks:** ### 1. Calculate the Gradient \( \nabla f \) Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \). \[ \nabla f = \boxed{\hspace{100pt}} \] ### 2. Calculate the Derivative \( \mathbf{r}'(t) \) Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \). \[ \mathbf{r}'(t) = \boxed{\hspace{100pt}} \] ### 3. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = 1.9 \) Use decimal notation. Give your answer to three decimal places. \[ \frac{d}{dt} f(\mathbf{r}(1.9)) = \boxed{\hspace{100pt}} \] ### 4. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = -1.5 \) Use decimal notation. Give your answer to three decimal places. \[ \frac{d}{dt} f(\mathbf{r}(-1.5)) = \boxed{\hspace{100pt}} \] ### Explanation: #### Gradient \(\nabla f\): The gradient \( \nabla f \) of a function \( f(x, y) \) provides the partial derivatives with respect to \( x \) and \( y \). It is a vector field indicating the direction of the steepest ascent. #### Derivative \(\mathbf{r}'(t)\): The derivative \(\mathbf{r}'(t)\) is computed component-wise for the vector function \(\mathbf{r}(t)\). #### Chain Rule for Paths: The chain rule for paths is used to differentiate a composite function. If \( \mathbf{r}(t) \) defines a path in space and \(
### Mathematical Analysis and Chain Rule Application **Given Functions:** 1. Let \( f(x, y) = xy^2 \) 2. Let \( \mathbf{r}(t) = \left\langle \frac{1}{2} t, 2t^3 \right\rangle \) **Tasks:** ### 1. Calculate the Gradient \( \nabla f \) Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \). \[ \nabla f = \boxed{\hspace{100pt}} \] ### 2. Calculate the Derivative \( \mathbf{r}'(t) \) Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \). \[ \mathbf{r}'(t) = \boxed{\hspace{100pt}} \] ### 3. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = 1.9 \) Use decimal notation. Give your answer to three decimal places. \[ \frac{d}{dt} f(\mathbf{r}(1.9)) = \boxed{\hspace{100pt}} \] ### 4. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = -1.5 \) Use decimal notation. Give your answer to three decimal places. \[ \frac{d}{dt} f(\mathbf{r}(-1.5)) = \boxed{\hspace{100pt}} \] ### Explanation: #### Gradient \(\nabla f\): The gradient \( \nabla f \) of a function \( f(x, y) \) provides the partial derivatives with respect to \( x \) and \( y \). It is a vector field indicating the direction of the steepest ascent. #### Derivative \(\mathbf{r}'(t)\): The derivative \(\mathbf{r}'(t)\) is computed component-wise for the vector function \(\mathbf{r}(t)\). #### Chain Rule for Paths: The chain rule for paths is used to differentiate a composite function. If \( \mathbf{r}(t) \) defines a path in space and \(
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 12CR
Related questions
Question
![### Mathematical Analysis and Chain Rule Application
**Given Functions:**
1. Let \( f(x, y) = xy^2 \)
2. Let \( \mathbf{r}(t) = \left\langle \frac{1}{2} t, 2t^3 \right\rangle \)
**Tasks:**
### 1. Calculate the Gradient \( \nabla f \)
Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \).
\[ \nabla f = \boxed{\hspace{100pt}} \]
### 2. Calculate the Derivative \( \mathbf{r}'(t) \)
Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \).
\[ \mathbf{r}'(t) = \boxed{\hspace{100pt}} \]
### 3. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = 1.9 \)
Use decimal notation. Give your answer to three decimal places.
\[ \frac{d}{dt} f(\mathbf{r}(1.9)) = \boxed{\hspace{100pt}} \]
### 4. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = -1.5 \)
Use decimal notation. Give your answer to three decimal places.
\[ \frac{d}{dt} f(\mathbf{r}(-1.5)) = \boxed{\hspace{100pt}} \]
### Explanation:
#### Gradient \(\nabla f\):
The gradient \( \nabla f \) of a function \( f(x, y) \) provides the partial derivatives with respect to \( x \) and \( y \). It is a vector field indicating the direction of the steepest ascent.
#### Derivative \(\mathbf{r}'(t)\):
The derivative \(\mathbf{r}'(t)\) is computed component-wise for the vector function \(\mathbf{r}(t)\).
#### Chain Rule for Paths:
The chain rule for paths is used to differentiate a composite function. If \( \mathbf{r}(t) \) defines a path in space and \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faccc7427-2c55-44a5-b68e-19581b7568be%2F054b7458-a2a6-4721-9607-e753d3d234f0%2Ff5cxteap_processed.png&w=3840&q=75)
Transcribed Image Text:### Mathematical Analysis and Chain Rule Application
**Given Functions:**
1. Let \( f(x, y) = xy^2 \)
2. Let \( \mathbf{r}(t) = \left\langle \frac{1}{2} t, 2t^3 \right\rangle \)
**Tasks:**
### 1. Calculate the Gradient \( \nabla f \)
Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \).
\[ \nabla f = \boxed{\hspace{100pt}} \]
### 2. Calculate the Derivative \( \mathbf{r}'(t) \)
Use symbolic notation and fractions where needed. Express numbers in exact form. Give your answer in the form \( (*,*) \).
\[ \mathbf{r}'(t) = \boxed{\hspace{100pt}} \]
### 3. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = 1.9 \)
Use decimal notation. Give your answer to three decimal places.
\[ \frac{d}{dt} f(\mathbf{r}(1.9)) = \boxed{\hspace{100pt}} \]
### 4. Using the Chain Rule for Paths to evaluate \( \frac{d}{dt} f(\mathbf{r}(t)) \) at \( t = -1.5 \)
Use decimal notation. Give your answer to three decimal places.
\[ \frac{d}{dt} f(\mathbf{r}(-1.5)) = \boxed{\hspace{100pt}} \]
### Explanation:
#### Gradient \(\nabla f\):
The gradient \( \nabla f \) of a function \( f(x, y) \) provides the partial derivatives with respect to \( x \) and \( y \). It is a vector field indicating the direction of the steepest ascent.
#### Derivative \(\mathbf{r}'(t)\):
The derivative \(\mathbf{r}'(t)\) is computed component-wise for the vector function \(\mathbf{r}(t)\).
#### Chain Rule for Paths:
The chain rule for paths is used to differentiate a composite function. If \( \mathbf{r}(t) \) defines a path in space and \(
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage