# Thread: calculus with exponents

1. ## calculus with exponents

I can't get a clear graph on here, but you can graph it on a graphing calculator, the graph of the curve with equation: y=e^(2x)cosx
On this graph, P(a,b) is labelled in the first quadrant as the graph is increasing.

a) Show that the first derivative=e^(2x)(2cosx-sinx)
b) Find second derivative

At the point P(a,b), the second derivative = 0.
c) Use results from a) and b) to prove that
i) tan a = 3/4
ii) the gradient of the curve at P is e^(2a)

Ok, so I can do a) and b), the second derivative is e^(2x)(3cosx-4sinx) and this is correct. I can prove c)i), but I dont know how to prove ii).

If I substitute 'a' into the equation of the derivative, then it's: e^(2a)(2cosa-sina), how is the answer e^(2a)???

2. Originally Posted by bhuang
I can't get a clear graph on here, but you can graph it on a graphing calculator, the graph of the curve with equation: y=e^(2x)cosx
On this graph, P(a,b) is labelled in the first quadrant as the graph is increasing.

a) Show that the first derivative=e^(2x)(2cosx-sinx)
b) Find second derivative

At the point P(a,b), the second derivative = 0.
c) Use results from a) and b) to prove that
i) tan a = 3/4
ii) the gradient of the curve at P is e^(2a)

Ok, so I can do a) and b), the second derivative is e^(2x)(3cosx-4sinx) and this is correct. I can prove c)i), but I dont know how to prove ii).

If I substitute 'a' into the equation of the derivative, then it's: e^(2a)(2cosa-sina), how is the answer e^(2a)???
you know $\tan{a} = \frac{3}{4}$ ... so, what would be the values of $\sin{a}$ and $\cos{a}$ ? ... and how will those values determine the value of $e^{2a}(2\cos{a}-\sin{a})$ ?
The text passage contains a calculus problem with multiple parts, including questions and partial answers. The user is asking for help with a specific part of the problem.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Thread: calculus with exponents

1. ## calculus with exponents

I can't get a clear graph on here, but you can graph it on a graphing calculator, the graph of the curve with equation: y=e^(2x)cosx
On this graph, P(a,b) is labelled in the first quadrant as the graph is increasing.

a) Show that the first derivative=e^(2x)(2cosx-sinx)
b) Find second derivative

At the point P(a,b), the second derivative = 0.
c) Use results from a) and b) to prove that
i) tan a = 3/4
ii) the gradient of the curve at P is e^(2a)

Ok, so I can do a) and b), the second derivative is e^(2x)(3cosx-4sinx) and this is correct. I can prove c)i), but I dont know how to prove ii).

If I substitute 'a' into the equation of the derivative, then it's: e^(2a)(2cosa-sina), how is the answer e^(2a)???

2. Originally Posted by bhuang
I can't get a clear graph on here, but you can graph it on a graphing calculator, the graph of the curve with equation: y=e^(2x)cosx
On this graph, P(a,b) is labelled in the first quadrant as the graph is increasing.

a) Show that the first derivative=e^(2x)(2cosx-sinx)
b) Find second derivative

At the point P(a,b), the second derivative = 0.
c) Use results from a) and b) to prove that
i) tan a = 3/4
ii) the gradient of the curve at P is e^(2a)

Ok, so I can do a) and b), the second derivative is e^(2x)(3cosx-4sinx) and this is correct. I can prove c)i), but I dont know how to prove ii).

If I substitute 'a' into the equation of the derivative, then it's: e^(2a)(2cosa-sina), how is the answer e^(2a)???
you know $\tan{a} = \frac{3}{4}$ ... so, what would be the values of $\sin{a}$ and $\cos{a}$ ? ... and how will those values determine the value of $e^{2a}(2\cos{a}-\sin{a})$ ?
user
The text passage contains a calculus problem with multiple parts, including questions and partial answers. The user is asking for help with a specific part of the problem.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS